Terrestrial
MaKtetim
a•d
ltmospheric Electricity
Vo•,uu•:37
DECEMBER, 1932
A NEW
THEORY
BY S. Cuxv3t^x
OF MAGNETIC
No. 4
STORMS
.•XD V. C. A. F•x•o
Px•'r I---T•F• xxxx•xu vuxsv. (Continued)*
9. Thevariation
ofpress.ztre,
density,
andtemperature
near
t)•ehead
ofthe
stream
By V. C. A. FF•aV, taO
9.1•The
examination of the conditionsin the gas near the head
of the streamshould,to be satisfactory,be basedon a kinetic theory
treatment. Asyet wehavenotsucceeded
in developing
sucha treatment,
andare,for the timebeing,forcedto adoptlessfundamental,
andthereforelesscertain,methods.In particularwe nowassume
that the value
of R,,/l,at the frontboundary
of the stream
is at leastof orderunity,
on the groundthat if it becomes
lessthanunitythe electrons
will be
unableto continuetheir advance(with the ions)into the field.
If R,/I• is of•order
unityat theboundary,
andexceeds
unityin the
stream,
theelectrical
conductivity
evenin theshielding
layerwillbeof
the sameorderof magnitude
asthe full electronic
(cr•)o
of õ8.1. Hence
the thickness
of the shielding
layerwill besimilarto that estimated
in
õ8.2.
'
At Z earth-radii from the Earth's center, R•/l•**=4XlO'nNZ•
when T--6000ø. Thus the assumed
conditionrequiresthat, at the
front of the streamN is at leastof the order2 X 10TM/Z
a ionsper cc,or
(2X10•0•
Za)(T/'6000)
•/'2if T differs
from6000
ø.
At distances
Z = 5 and2, andfor T=6000ø, thecorresponding
order
of N is 2X 10sand10ø;thesevalues
exceed
whatseems
to betheprobable
valueof N in the streamin the neighborhood
of the Earth,apartfrom
thedisturbance
of thestreamby theEarth'smagnetic
field. According
toõ8thedensity
nearthefrontofthestream
canberaised
tothisvalue
bythein-pouring
matter
frombehind,
asthefrontlayerisretarded.
Theproportionate
increase
maybeless
thanwasthere
calculated,
however,onaccount
of thebackward
pressure-gradient
accompanying
the
increase
of density--particularly
if thecompression
of thestream
of
gasraises
thetemperature
asintheadiabatic
compression
ofanordinary
*Continued
fromthisJournal,
S6,77-97and171-186
(1931),
37,147-156.
**Here,
asinõ8,it must
beremembered
thatReisnotthe$•firal
radius
which
ttteelectrons
would
execute
inthemagnetic
field
withtheir
velocity
of10• cmpersec.;
it would
bemore
accurate
(butperttaps
less
simple)
toreplace
Re/le
byYello
e,where
•o
e(--ell/me)
isthefrequency
ofsl)iralling
oftheelectrons
and
ve( --Cell
e)thefrequency
ofcollisions,
where
t5'
eisthemean
thermal
velocity
ofthe
electrons,
421
[Plate 3]
422
S. CHAPMAN
AND
V. C. A.
FERRARO
[VOL. 37, No. 41
gas. The extent to which these factors should be considereddepends
on the frequency of collisions,and therefore on the density and temperature
of the stream.
A calculation bearing on these points will now be given in brief.
In the first place it will be supposedthat the conditionsare isothermal,
and afterwards (õ9.9) the modification due to variation of temperature
will be considered. The analysis is an extension of that of {}8, and is
limited by the same approximations as were there adopted to lessen
the mathematical difficulties. No accountis taken of any flow of matter
transverse
to the stream.
9.2--Using the coordinates of õ8.4, the equation of motion of an
elementat the point (0, 0, z) will be
(99)
•= F- 0__p
Oz
which differs from (57) by addition of the term -Op,/Oz. Assumingfor
the present that the conditionsremain isothermal, the pressurewill be
given by
(100)
p=kNT=Rp
where (approximately)
(101)
R = (2/3) C •
C being the mean thermal velocity of the positive ions. For 7'=6000 ø,
the valuesof R for hydrogenand calcium ionsare respectively
(102)
Ru=9.6X10 n and Rea=2'4X10 TM
Substituting (65) in the equation of motion (99), and proceedingas in
õ8, we deducean equation of the form
=0
where
oQ
) o'q
(03)
whilst •, (whose form need not be considered here) vanishes identically
at the boundary z= z0,or d =0; it is therefore small for sufficiently small
values of d, and hence, as in õ8, an approximate solution of equation
(99), valid near the boundary, is a solution of the equation 4q=O.
If we
set
this equation can be written
(105)
O=Q
u,•-•
oQ- (,/g)Q=
- l•ff
a d• n
,
the solution of which can be obtained in terms of integrals analogousto
(80). These cannot be evaluated in finite terms, and here it seemssuffi-
A NEW
THEORi
•' OF'.,?I,œAGNETiC
STO?MS
423
cient to replace? by the approximate form (valid near the boundary)
e-*a. The general solution of (105) is then
(106)
Q=Q•
where Q•, Q• are arbitrary functionsof the time t, and
(07)
One condition to be satisfiedis that Qmust tend to zeroas (z-•)-•-
•
(cf. •8.4, p. 152); we cannotapply this conditiondirectly to (106) because(105) is only valid for smallvaluesof d. But sincewe couldregard
(106) as a solutionof the equation(99) providedthat F wereincreasedby
the amount • •,, a solutionof this equationwhich least deviates from
the actual solution is that for which Q•= 0, becausethis makes•, smallest.
Hence we choose the solution
(108)
Q=Q• e
To completethe solutionwe requireanotherconditionat the boundary of the streamd=0; as we havesaidin •9.1, this can only be determinedby a kinetictheorytreatment,in the absence
of whichit becomesnecessary
to assumesucha conditionas seemslikely to approximate to the truth.
The forceactingon the ionsis madeup of the electromagnetic
force
F and the pressure
gradient-Op/Oz;nearthe front boundaryof the
streamit seemsimprobablethat the latter can numerJca!ly
exceedF.
Thus the ratio
o= (-
op
/
seemslikely to be a fractioh numericallylessthan 1. Moreover, since
the density'
is likelyto decrease
aswe moveinwardsfromtheboundary,
0 is unlikelyto benegative.Its variationisthuslikelyto beslowasthe
stream advancesinto the field, and for this reasonwe shall treat it as
constant. This is the assumption
we makeand it will be seenlater
(õ9.5)that it is notinconsistent
with theassumption
that(R•/l,)a-ois
not less than unity.
The above condition can be written in the form
(109)
WQ
oa• = 0t5
whered=0, and 0 is a positix;e
non-dimensional
numberof orderunity.
Applying
thiscondition
at d=0 to theequation
(108),it canbeshown
that to leadto theequation
of motionof thefrontof thestream,which
is approximatelygivenby
(110)
where
g= R•gtS/(1
424
S. CHAPMAN
AND.
V. C. A. FERRARO
[voL. $7, No. 41
At large distancesfrom the Earth, where g ?,<•)•/2< < 1, the equation of motion becomesapproximately
o,
z0= This means that the density is approximately uniform throughout the
stream during the early stagesof the motion--and this is clearly true
provided g=l.
The equation (112) can also be obtained if we make
R-• ooand hence T-•oo, which suggeststhat if the temperature of the
stream is infinitely large there can be no heaping up of matter.
9.3--If
we write
(113)
(110) becomes
(• •4)
g = R'y/Z
The solutionof this equation, which is similar in some respectsto (32)
in õ7, can be discussedby the method given there; and it may be remarked here that both these equations have no solutions such that
v--•0 as Z-•,0: The (at first sight surprising)absenceof such solutions
means that the front of the stream would be brought to rest before
reaching the originwa result which is unlikely. The explanation is to
be sought in the admittedly approximate methods used either in formulating the simplifiedproblem (õ7) or in the mathematicalapproximations
used (õõ8, 9). In õ7 the equation (32) is no longer valid when the velocity of the stream is reduced to a small fraction of its initial value,
since it takes no account of the thickening of the current layer, which
increaseslike 1/w (cf. õ8.2). In the problem consideredin {}8and in the
present case the result is probably due to the neglected term •P=. The
solutions of these equations, nevertheless, give at least an indication
of the orders of magnitude, since•2 vanishesat the boundary and at an
infinite distance, and does not exceed the value of the terms retained in
the equation of motion at the boundary, except possibly when w is
reduced considerably.
9.4--The equation (114) has been solved partly analytically and
partly numerically down to the region in which w/woo first begins to
decreaserapidly; the solution which is of importance for our application
has the approximate form
M being the magnetic moment of the Earth. Inserting numerical values
for the various constants, taking Woo= 108cms per sec., T=6000 ø, this
becomes
(116)
g= (K./A'-'Noo)
•/3 2'8 >(101•:/(-Z) 2/3
and this solution is valid in the region
(117)
120
It is of interest to compare (116) with the correspondingvalue obtained in õõ7 and 8; there we find
g= }.(gœ'-'/"2z-p=o,w=o'•)•/2/(--Zo)S=
10• (AN=o)•/2(-Z) '2
.t
NEW
TttEORY
OF M. IGNET1C
SI¾*RMS
425
Hence the compression-intervalg varies much more slowly with
the distance -Z when accountis taken of the pressure-gradient,
as here,
and also it is very much larger; sucha result was to be expected,because
the matter near the head continuallytendsto spreaditself backwards
and thus to diminish the density and increasethe retardation of the
particles in the stream.
For reasonsstated in õ9.3 we cannot deduce from the solution of
(114) the motion of the front of the stream during the final stage when
the velocity of the stream is being reduced to zero; it seemsnot improbablethat this further decreasein velocity will be rapid at first, and
afterwards very slow, in a manner similar to that indicated by the
curvesin Figure 7, õ7.8. But during this last stage the flow of matter
along the lines of magnetic force may becomeof importance,and the
rise in the magnetic field producedby the stream may begin to decrease
as soon as the velocity of the front surfaceis reducedto, say, 1/10 of
its original value. The averageintensity of the initial phaseof a magnetic storm is about 303,,and, as was shown in õ7.9, this would be produced if the front of the stream came within 5 Earth-radii from the
Earth. From (117) it follows that A2N•/K must be of order 8X106;
sinceK is likely to be of order 1, this would mean that ASN•omust be
of order 8X10 •. For streamscomposedof hydrogenand calciumions
respectively,
the requisitedensities
of emissions
at the Sunwouldhave
to be about 4X10 n and 2)<10s respectively,taking account of the
geometrical
broadening
of thestream.Thevalueforhydrogen,
although
an overestimate,seemsmuch too high for the density of the stream at
the Sun's surface; the correspondingvalue for calcium appears more
satisfactory,
thoughit alsoseems
high. Buttheseestimates
aretentative,
and might be reducedby a factor of !0, or even 100, if a more accurate
examinationof the state of the gasnear the front of the streamcould
be made. The values of Noo, however,seem unlikely to be smaller
than !07 and 10* for H and Ca, becausetheseestimatescannotbe smaller
than thosededucedfrom the analysisgiven in õ8.
It seemsworth while pointingout that the requireddensitiesof
emissionfrom the Sun as here estimated differ by a factor as large as
103
, apparentlybecause
both the thermalmolecular
velocityand the
retardation
of the particles
in a calcium-stream
areconsiderably
smaller
than those for a hydrogen-stream.
9.5---Vx;enext examine the distribution of matter within the stream.
To a first approximationwe find
(119)
Q=e-q•a
whereq•isnearlyequalto•/R, or _(•//g)•/2Zs;oninserting
numerical
valuesfor • andg from (113)and (116),we find
(2o)
½=0.079(•A.
The excess
densityis, by (75), proportional
to OQ/Odand henceto
e-e,a,andthegradient
of density
thusdepends
onq,. Thisfunction
increases
with -Z at a somewhat
rapidrateand,aswasto be expected,
is
largerthelargerthevalueof A, thatis,theheavier
theatom.On the
otherhand,it varieslikeNa 1/*,sothat theconcentration
of matteris
leastwhenthe initialdensityishighest--aresultwhichisotherwise
evi-
426
$.
CHAPMAN
AND
V. C. A.
FERRARO
[vot.. 3?. No. 41
dent since the retardation of the stream is smaller the larger Noo is.
Since • is also of order unity, (120) is approximately equal to 0.079
(A/2Voo)•/3/'(-Z) a/3. Thus, 99 per cent of the excessmassaccumulated
at the front will be contained in a layer 58 (2¾'oo/A)
1/3 (--Z) 8/3 cm
thick. At a distance -Z = 5 and for Noo= 8 X 106/A'2, this becomes
9.4/A km, or 9.4 km and 0.23 km in the caseof hydrogen- and calciumstreams, respectively. It is not surprising that these estimates considerablyexceedthat•of about 4 cm--given in õ8. The latter, however,
remains the estimated thicknessof the electric current-layer, calculated
from the kinetic-theory value of the conductivity; not much significance
can be attached to it, in view of the much larger value of the mean
free-path even at the head of the stream.
The excess density at the front of the stream is equal to
-Noog (c9Q//Od)•-•o,
by (75), or .gq•Noo. Using (116)and
(120)we
deduce
(121)
N=(N•:/A)
No• = 8 X 106,'A.2,
(122)
•/3 2XlOt•,,"(-Z) •ø/3 ions per cc or, taking
N=4.4XlOt•/A(-Z)
•ø/:•
ß
In õ9.1 we have seenthat for R•,/I• to be at least of order unity, it was
necessaryfor the density to be of the order of 2 X 10•ø/(-Z) * at least;
comparingthis with (122) it is seenthat the conditionwould be satisfied
both for H- and Ca-streams
at the head of the stream.
It is also satisfied
within the stream, because the density increases much more slowly
than the magnetic field as we move inwards from the surface, and the
difficulty with regard to this mentioned'in õ8.42 does not arise.
We wish to emphasize the uncertainty which is attached to this investigation, because of the assumptions and approximations involved;
but the resultsdo not seemto us improbable.
9.6-•According to (122) the density of the gas near the front of the
stream would be increased in the ratio 2.0XlO•ø/(A.Noo•)•/3 (-Z)•ø/3;
this, as was to be expected, is larger the rarer the stream and the
lighter the ions present, and except for very large valucs of No• (for
example, 10t• ions per cc), and at large distancesfrom the Earth, this
ratio will be large in comparisonwith unity. Taking, for example, as
before 2Vo•=8X10•/A z, the values of this ratio at distances from the
Earth equal to - Z = 5, 3, 2 would be 2 '6 X 104A, 1'4 X 10sA, 5 '5 X !0 • A,
respectively. For Ca-ions this is as large as
Such a large increaseof density would be accompaniedby a rise in
temperature if, by collisions,part of the kinetic energy of the stream is
converted into random motion. It is of importance to examine this
point in some detail since it considerablyaffects the estimates of the increasein density.
We find that, of the total kinetic energy transformed during the advance of the stream into the Earth's magnetic field, only a small fraction
reappearsas magnetic energy due to the magnetic effects produced by
the current-bearinglayer: Thus at a distanceof 5 Earth-radii from the
Earth, taking as usual Nm =8X10•/A •, w=(1/2)w•,
this fraction is
about 0.001 and 0.04 for the casesof streams composedrespectively of
hydrogen and calcium atoms. This suggests,therefore, that a large proportion of the kinetic energy of the stream becomestransformed into
heat-energy.
A NEW
THEORY
OF MAGNETIC
STORMS
427
The precedinganalysissuggeststhat in the layer containing99 per
cent of the excessmass accumulated,the variations in velocity from
point to point are small comparedwith the velocity of the front of the
stream (unlessthis is very small indeed,for example,about i0 a or 10't
cm per second). The wholelayer may thereforebe supposedto move
bodily with a velocity equal to that of the front of the stream. If this
velocity is reducedto a fractionf of its initial value woo(= 10a), a proportion 1--ff of the kinetic energyis convertedmainly into heat-energy.
If this is not dissipatedfrom the stream, the temperaturemust rise considerably. Thus, if C refers, as before, to the thermal velocity of the
positiveionsat the temperatureof 6000ø, the temperaturewill be raised
approximately
in the ratio {(1.--f)wo•+C} 2/C2. Hence,whenthe
velocity of the front of the stream is reduced by 106, 10*, 10a cm per
second,correspondingto (l--f) being ! 7100, 1/10, 1, the .temperature
would rise to 2 '4 X 10•, 4'2 X 10•, 4'2 X 10*økin the caseof hydrogen, and
tO 1 '7 X 10s, 1 '7 X 107, 1 '7 X 10•øk in the caseof calcium-streams. These
temperatures are extremely high, and unlessthe heat can be dissipated
away almost at once, our estimatesof the ratio of increaseof density must
be considerablyreduced. The only mechanismsby which the compressed
stream can lose heat seem to be those of radiation and ionization;these
will now be considered.
9.7--Radiation of heat in the presentcasemay take place by at least
three distinct processes' (i) Continuous emission, (ii) line emission
due to electron-captures,and (iii) line emissiondue to inelasticcollisions.
The emission due to the encounter between an ion and an electron,
giving rise to the spectra (i) and (ii), can be calculatedon the basisof
Kramers' theory of absorption,for it has been shown by E. A. Milne
that this theoryholdsevenin the caseof a gassuchas the photosphere
of
the Sun. The resultsobtained by Milne are discussedby A. S. Eddington* who finds that for singly-ionizedcalcium-atomsat a temperature
of 6000ø, and at a pressureof !00 dynes per cms, the mass-absorption
or emission-coefficient,k', is about 150 c.g.s. This value seems to be
uncertain to the extent of a factor 10. Sincek'a NIT ?/2,for a gaswhose
densityis N and temperatureT, the appropriatevalue of k' is
(123)
k'= 1'37 NT -7/2
The emissionper gm per secondof a gasat temperatureT is givenby
(124)
4k'• T 4
where
cr(=5'32XI0L•)heredenotes
Stefan's
constant.
Using
thevalue
of k' givenby (123) the emission
per cc persecondis
(125)
j =4'7 X 10-?8NST1/2ergs
This expression
showsthat the variation of the emissionwith temperatureis onlyslight,the densitybeinga far moreimportantfactor.
Some of the emitted radiation will be re-absorbedby the material
and shouldbe deductedfrom (125) to get the net emission;and it seems
likely that the line emission
may be cut off altogetherby line absorption
leavingonly the continuous
emission;
this mightreducethe valueof.i
*The internal constitution of the stars, 355-359 (1026).
428
S. CIfAPMAN
AND
1/'. C. A.
FERRARO
[VOL. 37, No. 41
by a factor of 10 or more, except, of course,in the surface-layers. Thus
by using the expression(125) we shall over estimate the emission.
We shall continue to use the estimates for the density and velocity
at the head of the stream deducedin the previous section for a calciumstream.
Considering a column drawn in the stream 1 sq. cm. cross-section
with its base in the surface of the stream
and whose axis coincides with
the axis of the stream, we have seen that the excess mass would be confined to a length of this column of about 104cm with a mean densit3, of
about 10•ø ions per cc when the front of the stream is at a distance of
about 5 Earth-radii. Using thesevalues and (125), and multiplying by
4,r, it appears that the total emissionfrom this length of the column
cannot exceed 0.59 T 1/2 ergs per second. At the temperature 6000ø
this becomesequal to 46 ergs per second.
The rate at which kinetic energy is being transformed in this column
is approximately
Am•3
•1 Noo
(126)
where} (=woo-}0) is therelativevelocityof theparticles
at thebackof
the stream
relative
to the front
surface.
For a calcium-stream
when
.
--Z=5, .g,-•woo,Noo=4000, and substituting these values in (126) we
find the rate at which kinetic energy is being transformed in the column
is of the order of 10• ergsper second. This is 2000 times the rate of radiation of heat found above; this conclusion,that emissionat the rate (125)
cannot disposeof the transformed kinetic energy without a rise of temperature above 6000ø, can be drawn less definitely but more simply, in
the following way:
9.8---In order that matter may be heapedup at the front of the stream,
the velocity in front must be decreasedby an amount at least equal to the
velocity of thermal agitation of the ions per second;that is, for a calciumstream, by 105cm/sec'2. Otherwise the matter would continually spread
backwardswith approximately this velocity as fast as it is heaped up.
The region where this retardation will first take place depends on the
initial density Noo of the stream. A rough calculation indicates that,
for values of Noo lying between 10 and 1000, the front of the stream begins to be retarded by the magnetic field when within 1000 Earth-radii
from the Earth. Taking woo= 108, the front of the stream would be
brought to rest in a time r = 1000seconds,when it would have comewithin
a distance
of about 6 Earth-radii.
The total mass accumulated
would
be of the order of (!/2) Noo woor ions/cm'•, and if N=o= 1000, this is
10•4ions/cm'". It is of the sameorder as the excessmassper sq. cm. found
previously.
We cannot estimate in this way the thickness of the layer in which
the greatest accumulation of matter occurs, but we can hardly suppose
this to be smaller than the value of about 4 cm deduced in õ8. Taking
this to be of order of the thickness of the layer, the density at the front
of the stream may be as great as 10•:•ions/cc. Using this value of N, and
taking the length of the column to be 4 cm instead of 104cm (as was
donein õ9.7)the emissionper sq. cm. of the surfacewould be 10• ergs/second instead of 46 ergs/second. Even this overestimate is less than the
rate at which kinetic energyis transformed,namely, 10aergs;/second;
thus
.t NEII'
TtlEORY
()F 2I. tGNETIC
N?'OR.1L¾
429
such radiation seemsinsufficientto prevent the temperature of the gas
rising above 6000ø.
9.9---Emission by inelastic collisionswill be to some extent counteractedby superelastic
collisions;
but if the densityis low, for exampleless
than 10TM
ions/cc, the latter will be negligible;in fact, an inelasticimpact
can be effectivein emitting the energyof collisiononly if the colliding
corpuscles(two ions or an ion and an electron)approachwithin a dis-
tancecomparable
with the radiusof the atom, 10-scm. The free-pathis
thereforeof the orderof 10t•,/Ncm, and the time of describing
this path
is 10t•/(NC) seconds,C beingthe relativespeedof the collidingpair of
particles. Taking for exampleN=10 •øions/cc,C= 10s cm/second,
the
above time becomes10-a second. The average life of an excited atom
is of the order 10-8second,so that as.longas the densityremainsbelow
10•aions/cc,the superelastic
collisionsmay be neglected.
The ineleasticcollisionsof most importancewill be those between
ionsand ions,causingradiationor ionization,and in eithercasei
disposing
of someof the kineticenergyof the stream-motion
withoutaddingto the
thermalenergy. For theseprocesses
to occur,the collidingparticles
mustapproachto a distance
of atomicdimensions---say
3 X 10-scm,corresponding
to a meanfree-pathof about ,,,
•n• /N cm. The thicknessof
the layer,if, as calculated,
thiscontains10TM
ions/era'"
column,will also
be 101•,/N,
sothat in passing
throughthelayerthe incoming
ionsfrom
behindwill onlymakeaboutonecollision.Evenif thisis alwayseffective in ionizingthe particlecollidedwith,the energysolost,of theorder
!0 volts or 2•10 -• erg (say) is onlya smallfractionof the energyof
motionof the incomingion relativeto the layer (e.g., if this relative
velocityis (1/'2)• l0scm/second,
theexcess
energy
fora Ca-ionisoforder
10-=erg). Thusif theincoming
ionis to dispose
of itssurplus
energyin
the layer,it mustbe deflected
(mainlyby the magnetic
field)andmake
manysubsequent
ionizing
collisions.
Meanwhile
itskineticenergy,.being
no longermainlyin thestream-direction,
countsin partasthermalenergyof thelayer. Thusit seems
likelythatthetemperature
of thelayer
mustbe raised,thoughionizingcollisions
maydispose
of an appreciable
partof thiskineticenergy.To theextentthatsucha riseoccurs,
the
difficulty
isenhanced
asregards
thevalueof Noorequired
forthestream
to comewithin5 radii of the Earth (consistently
with the maintenance
of
Re•t• at the front of the stream).
--•or example,
suppose
•/'is increased
from6000
øto 100,000
ø (10•'C').
SinceRoe
T: (115)shows
thatg remains
unaltered
if we increase
N=oin
the sameratio, so that the estimateof A'2N=ogivenin õ9.4(namely
8X 10•) mustbeincreased
to about1'4• 10•. ForCa-atoms
thiswould
makeNoo=10•, corresponding
to a densits'
of emission
at the Sun
(allowing
fo,'geometrical
broadening)
ofabout5X 10:•,whichin thelight
ofpresent
expectations
seems
excessive.
•nfortunately
thefacts
astothe
emission
at theSunareasobscure
astheworking
outof theirterrestrial
consequences
is difficult.
In concluding
thissection
wewishagain
tostatethatweareconscious
ofmany
unsatisfactory
features
inourpresent
treatment,
which
exposes,
ratherthanovercomes,
the theoretical
difficulties
of theproblem.We
intend,ho.wever,
to makefurtherefforts
aftera solution,
alongany
avenues that may seem open.
(To becontint•ed)