A THEORY OF THE PROPAGATION OF ELECTROMAGNETIC WAVES IN A MAGNETICALLY ACTIVE MEDIUM

• Document Type: CREST
• Collection: General CIA Records
• Document Number (FOIA) /ESDN (CREST): CIA-RDP82-00039R000200100034-2
• Release Decision: RIPPUB
• Original Classification: R
• Document Page Count: 29
• Document Creation Date: December 22, 2016
• Document Release Date: May 8, 2012
• Sequence Number: 34
• Publication Date: August 6, 1952
• Content Type: REPORT

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A TheorY_of the ?z
agation of E ectrom ,one is
in a MaEneticafly Actiee Med
Zhurnal EksperimentalofOY i Teoreticheskoy Fiziki,
Vol XVIII, No b, pp 487.501, Jun 4$, Russian Mo per,
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STAT
STAT':;'
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A TliitlC C)F THE PI OPAQA2WN O CTROMACNE'ITC WAVES IN A NACNB'1iCALL1
ACTIVE 1'/1EDI
V. L. Ginzburg
The article is devoted to devolopment of a theory of the prat
pagation cf e1ectromagnetiC waves in a magnoticaily active mediwr
(in the ionosphere with p~,sidcaration of the effect oi' the earth's
magnetic field). An _approxUmate solution is prformed f'or` the equa-
Lions of propagation in the region of reflection o; a wave from a
wui r{;
layer and at' the beginning of a iayeIwOCO geometrical optics is
not applzcable- I1hc; ef.~ect of trip1in_ of signals is considered
a
in an approximation which holds for small angles between the magnetic
field and the direction of propagation. 'fhe problem of the a.
fh c
field is clarified, and by means of a kinetic"sequation method an ex'
pre s sion is obtained for the effective number of collisions, taking
into consideration the presence of the magnetic field.
14 A general consideration of the problem of the propagation
of electromagnetic waves in a magnetically active medium (in the
ionosphere with consideration of the effect of the earth's magnetic
field) has been made by us earlier (1) ( reference is henceforth
f
cited as I).
pe
with normal incidence of a wave on a' layer whose prom
rties depend only on height (coordinate x), the electromagnetic
field of the wave is subject to the equati.oris (See 1,
(2y
(iL)-(16):
STAT I.
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where
I = (i-v) .~ G1 GaI
,V
.
d , .,
The earth's magnetic Ia.e1.
p rectian off' ~a~^apagata.oi~ .~. the x
axis an~ xz and to form Via, th the d'~.
.- the angle In (i)?(2) is the
of: the wave and N( x) is the concentration of e:lectroIl5 in the layer.
e
In the approximate ~;eometrical opticS considered in T, th
a ~i.on o:t~ equation (1) iS as follows
s~.ut.
?()
s r U~
E .,,2 ',~ Ya/)Z,
soiution of the equation
th
2
e
is
where n
2
1
,
Cjp ~
- fl-
3
thus
2v C-)
__ ( L. )
r))L- (iV-U6ir tN+ft3
l+ui t} er~
-_ _ ___
-n i
((V)o$
4
v~
Lk
hick corxcs~pond to the upper and.. lower
the indexes i and 2, ~'
jflS in font of the root in (L.1) and
(L.2), characterize th.e type
I' .~Filiptl:~ry~~~ ~: E+i~ ?
; y, ~~ r Declassified in Part -Sanitized Co
((/V) - (i- v) z
- /- v)Vew
r~ ~uf (' v(i- y)
/// ,
Z{ ?) i assumed to lie in the
T
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0th//.4t I 7 ara'l. r"y II
of i.e., 2) or an we4rt (index
1) wave. - o1ution (3) rigorously satisfies the equatLon:
(Ahz) 'aE?>= C
. (pn/ ____
2n I I .~ ~
(5)
c2E (a/4 /dx )C1o) 0
~
1 Rf[
Without giving more detailed attention to the conditions of applic '
ability of . the approximation of geometrical optics (see x), we shall
note here only that in the case of a smooth ionosphere layer(l~2),
this approximation is inapplicable only when the inequality
cr>/2.yr') -- (C'/CJ5 not observed;
and when v4 0, i.e., at the very beginning of the layer.
(6)
The inequality (6) is not observed in the region of reflec'
Lion of the waves from the layer, where the index of refraction nl
is small and, in particular., when it is, equal to zero, Ordinarily
in such cases none-observance of the condition of applicability of
geometrical optics [ condition ( 6)] to one of the waves (
'i or, '~u t zr ial'~ ) this c edition 'c
the wave of the other type. However, wheni ~t 0, geometrical op-
tics is simultaneously inapplicable to waves of both types, and the
'ttriplingn effect occurs (see x),
In cases where geometrical optics is not applicable it is
necessary to find a rigorous or correct approximate solution of
system (l), which hitherto has not been done and to which the pre?
I
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nt ar ssentiai1 dovotad, In Naragraphs 2 and 3 tvesti'
Be~,rlc~e ~ e ~'
solution in the re @Qfl of relect'on o1 a wave
gation is made of the
from a :Layers In ~,ph L. brief co~1s~.derata,on is giver to the
~'ara~x
?1 _ ~ Paragraph
on v' 0)
be innin ; o the layer ( the x i
solution at the
e t~tripling" e 'iect. In ?arsgr; h 6 the d scuss an
~ is devotad to th /~
~y1#: M' I r
of the a: O field and the of f'ec Live number off' coi '
invo1vos problems
ls.s in the presence of a magnetic Lieid?
~.a~~s in the ionosphere 2, In thc~ absQncE; of a magnetic 'ie1d,, when u = 0, equation
(1)
With the excep~r,quencies very close to the critical free'
~
Wa.~t~.on of i~
quency, the layer in the region or reflection, where geometrical op~
)~
i~e?~
tics is inapplicable, may be replaced by a linear form (2'3it may be as sumed that v ax. In this case the changc~ in phase of
the wave as the result of its passage into to layer and reflection
. , (2'L')
From it is equal to
~I
(=L
'r C, 4J z (8)
0
The solution of equation (7) for a liner layer is expressed by ?the
integral or in Bessrel functions of the order ?1/3
In the presence of a magnetic field the equatians Of (1) re-
an equation of the type of (rr) in two particular cases ..w
duce to ,
/2 ) propagation --
and transverse (
with longitudinal (c~ =0)
--
when
wO 1 h -
C
assumes the form:
( % /a)+(&f~/a2)n El 0)
tl~?~nS ?B8I&? y~Fn tit''lr'i0f rS`rsr~4 ~l1ly~ 1~nf
ti v:Y I i??a1 N'.~ 6'~V~~~ QQ ~1Lt~''ti t y~+ r~y~tft 1,1, !~N G '.A a 1 J~ r~ 1 ~~.k: ~, ali
tr,;l
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-Cc
V)/()-t-V)
rV\
In the case of (9) ~.t the variables F,1,
ter (1) assumes the form:
d.? Fi
dx2
(10)
i , the sysw
-( _
With transverse propagation [ the case of (10) ] wry have;
dE _ C f-(
-- - -~----
-~ l ~-~ M v
K~ C
#(t)E
~2 C
(12)
a,?.. 6k.
We see that in both cases the var iablesAsepara^ and the
sy~3 tern of ( 1) actually reduces to Jtwo ecfuations of type (7) Hence
it follo~rs that with o0 and o~ for the phase , and also for the :field itself, there occur results which are correct in the
w
in the absence'of' a field, of course, with the su bstitution of n
for n l 2
for examA]?e, for C formula (8) is correct wit(i.e., 0 (
For the equations of (U) and the second of the equations
n = n1, 2 ).
of (12) this assertion is obvious, since they are id.e with
in the sense that v enters into them linearly. Into the .C'irst of the
equations of (12) v enters in more complex fashion, and therefore,
even for a linear layer
v=ax+b (13)
and n is a non'-linear function of x. However, for correctness of
1
formula (8) and of the entire solution for the linear layer it is
merely necessary that n may be replaced by a linear expression only
1
geometrical optics is inapplicable Se1ec~
in the region where
2
tin the origin of the coordinates at the point nl 0 (assuming that
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w~1 iett
Thon, on condit:lan that
and the first equation of (12) reduces to
function of x.
with nh as a linear
Appla.cati.on o:L' the condition of (6) to (lo) show, that ,eo
~' 111e tw
rival optics is applicable if
(17)
'1'lxus, if the conditions of (1S) and (17) are si,multan u
eausl.y ful.?
filled, than formula ($) and the others ara
c, correct also as applied
to the first equation of (12)e For the !iayer, for example, a''''l0 5
6' 103 and when l ? u~ ~
1, cond1t1on (15) denotes that
x 106 W 1.07, and c ondi t.L on ~
(17) that x)7 10 , ire,, both inequalities
are completely compatible,
3, The b reaking down of the e qua ti ons of ( l ) in to two ind
epen
dent equations of the second order takes place only in the to indi c
aced instances, o = 0 and : , t ified b
~ by the fact that in their
case the; wave~;riape is not a function of x (i`e.
R w con, t) , In
general then, R = f~('x), the e quations do not break down, and may be
reduced only to one equation oV the fourth order ~ ry
ox .Ear ~ or E ,
they are extremely complex, Tkzez"efare it is expedient here to resort.
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lutiof. With ?the r xoeptiOf of he II trip1ixlll
to an approXimatac~ sQ
rr~g.ch 1i13 ~ values of o~ in the re~;~.on of rc~f~,oo"
~.an~ wha.c~b~ , toy s ma~~.
ta,on f the waves rarn the layer, gepmet:xioa1 optics is .inappll.aabit
o ,r
only fi:or a wave of the one type, and i~ appl.a.Gabl,a for a wavy of the
--^ae r1a. pbv' OUs 1.Y t1 p a 1.n With the tact that a wave of type
~'gnd ~~'pQ~ 'hu '.~
2 is ref1ectd near the point, . n22 ~ q (i.e. v20 : 1), and a wawa of
2
te of the pq~.n~,s = 0 i.~'., va,0~ " 1 ? 2).
1 is reflected at one
Ule red. ore y in the z e @on of e f'1.ec ti on the waves of both types are
independent to a good ap.roximata.on (s?e x i'or more details), and
.~
may b Px subject to
?1sec;ted that each of the waves is approxima?tciT
e
an wa,th n2 2 substaitutcd {'or n2 ,~us?t as 'talces
equata. on of type ( ~. )
place in the cases of (9) and (10). This assumption proves to be
correct
C,F.7+t.
i r we to the variables
F= L ? ~ ,
?~ z
c ;
~ ~dx, ~lI w d2lt~l~~~)
t,ktpn the equata.ons of (l.) as,.~uarie the f'orra (R' dk~
; F? _
I
x 2P QI
(19)
Talctnp' into consideration that by virtue of (2) and ()
?.?..w ...~ .. ."..".. """..
+ B iC(R2 1) , 2 B_ A j G(R2
2 21
(20)
of the individual. of the equ.atiol
and eval.ua~ta.n~ the . order
(l,g )' it a.s natural to seek a solution; of this system in the form:
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(F1)\xne`trira1
p?
1a w
G,n f d,rl f ?-, ,r ..., ,, .;~?~ ,. :.. u- I w:a, t Irv.. I,.~:...,.I .;. .,. J;;:, ,... ~,. ..:r.. ,: ~: .. ,.I I .C , ,1 al ~.IIF,7. :. ~I~.
"uF Declassified in Part - Sanitized Copy Approved for Release 2012/05/08: CIA-RDP82-00039R000200100034 2 '"' , `{ "k ,,`'"
~sw-r~..,.,11 ~, r+` a.J,.~r.. ,..!,.:.,I.., sr.,.. !, i.),.,~1 ,.;e a. ~..;...._ .__.. .. _.._... _.. _._... .._._ _. .. .._.__ . .. .. _... ____ _. _ _.. __. .. _. __. ..... .. ,. .,, .., ?~.:..r,..,.,... ~~..., 1.u, . ,...,!'',..: ~'~~ .?. mr'^,: ~I ,~+r.~c,k?rh8,
 Declassified in Part -Sanitized Copy Approved for Release 2012/05/08 :CIA-RDP82-000398000200100034-2
ap~~.c~ (d~I+~o)~dx~) ,.`((~~/c~}n~.~~~~) ~ (C~/c)n~~~~~0)~ a~~c~ ~.n ~ha
:~
rp . ion ~a~ ~ 0 ~tha d~v:~~t~.ar~ :('xom tha ~ppra~~n~~t~.on off' ~?omptr~ca~~
~ ~
opt~~~ ~.s off;' tha a~~dax o;C ~tha ~a.a~,d rr~~~a.tucle ~.i;~a~.~,~,,o~,
(d'" F ~ yc
sec the so~.u~tl.ans ~.~ ~ ~"~~) ~ :~ur~th~~^ d,~'(0)/dx ~^ (~ '~~ )n ~'(G)
( ~ ~ ~ ~. ~
(z~ ~, )~'~ ), whence ex:pxo~sion (~0)~ is dexa.ved, ~'
va]~ue off' ~'(x) from (;,0) in~c- (~2f3) ~a.t may be 'seen ~;hdt
w
i~ v3,0~' ~
Tn tl~e r~~;~.on (which is in~~restin~ .to us) a~aund the point
~ ~? ,~ ~ ~ ~~ ~ i s the aa~.utian o~ r'(a) is a and
n~ ~ 0 clv/r~~~l,,., a,. ~~ lg .and ~,h.~ ,~ ~
0
a ro~.mation the mare so i~ that actu~J.ly in (3~.) tYie ai~;n ~( may
pp ~
be substituted :Cor the siC~n
for a wave o~ tfpc 2 ire thq nei~hbarhood off' the point oi' xe-
:['~.ec;tion v = ~.g where n~ ~ Og an the other hands the appray~.mation
20 ~
which has been made is na~t va~.id, ~,rtua:l.~.yg as ~"ollows From (2) and
(~.~), ~.n the nex~;hbarhoad a~' 'this point 0 (~. v), /( ~ r,
,~~ 0) ~ ~ r v "'~'d~'~ 0) clx and in ~'(x) the prirLCipa~. /~'~,~~
}.
(1 ~ v)"~~'(0), `~'h.e~?e:Core, cond~.~.on (2~,) is na1; i'u~,fi].~.ed, and ,4~f
,f, ;~.
t11e entire approximation is inva~~.d,, a.nd thgx~q:~oxe spec]..a~, a~;tenta,on
mist be ~iv~n to the region v ~ ~.,
~n..this ,region we sh~,~.~, ~,et
v ~ ~. ~' aYg aX ~~ ~ e
~~`~~ d '~ , ~~ ( ~ ~, t
~~"~ ~ ~ ~,~ 'then with. ~ precision extending; ~;a
~II1t,~
aa~ss the s~~tem ~ (~i) assumes the ~nrm.
y~~~i+~t~cyy~,,' l' ~ ~,~~s:
'y~''!.~'""q?r?
(3~)
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rove or a ease
__ _ PY pp _ _ _ _ _01000342 '+~`.. ~ ~", ?~, !~~~, ~~ .`~~'~ , :~'~.",. ,,' ;~ (~.;~,~' ~.'~.~.~~?~;~~,~,
__.
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L CDS d' ~X
`~~~ 51r1~'d
0
~ ~ ~~~ ~ .~~,u2g~,n~~{ fax ..cog ~ .
A ~a~/sin a~ ~ 2
(~~)
eomc~r~~.ca~. op~~,cs and. on conda.~ion ~ha~
W~.~h the approx~.ma~~on o~
~.,
ax cos d~ /uNsa~n~ ~t ~ {~
?' ~ ~:'ac;~r, su~~~g~;s ~~he usa o:~ the aZ~proxa.ma~a.on
~~. ~~
;~' ~ ~~ ~ ~a~l
~~ ~
~~
C ~ ~ z
~, ~
~?~ ~~s
~X
(off
(3~)
AXE ~~ ~~x~(39)
:1.2 -M
where
~~ ~ ~ ~~' God al ~ GX
~X_
k~~x~~r~,ax~~
dye
~'o~ L~ ~-} we ha~re
~'' ~ ~'
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h a~ ~~uat~or~ (~9)
'~'~~ ~Q~,'~t~G
I~ ~~ ~ ~ ~
~ .~ ~ C ~ ~
.. ~ ~,~~ ~
J y
ar
a~~ ?~u~~~a~ ~~~)~
,, ~,s ~,h? s?~,u~~~.on ~~ ~~h~ horno~ena * ~~/~,
where ~'~rq
o~ (~,~.) wa mad' ~.at ~~ ~`
? valuata.on a:E the.. ma~na,~ude
n~ ~ ~, and that 1;l~c~rc~~'axa
n an :Eur tk~,o ~~ ~;ha t ~
tak~.nl~ an'~a aon~~.dsrat~
abl ' 1QSS ~ranr~~~r~ced than fox
a:E ~(Q~ o~ x ~~ aona~.d?~
th,e depandenc~' ~
~~~ -we have
~~ al~in~', ~~' ~,h~.s
nc~a.ta.an o~ (36~ ands s tra.ctly' ape
.
W~.th tho co of (3"~) ~~
", ~ the second a.nequal~.ty ''
a.s valid .also ~. ar ~.
conda.tion ,;
. ;E these ~,nequa],a.t~,es :Follows ram `;
? .the f~.rst o ?~
d,e~~ived from t ~ ~ ~ o~ ,
is
~~) ~ Thus ~ with the ' cond~.t~-ans `'
cand and; Pram equat~.an ~ ~ ~~?~ '~
the se ~ ~
a.s val~.d?
a ~~ r a~,irna t~. on ~ ..~~
the ~ ~ ~'
~ ~ and (3~) sata,s~'ied~ ~ ~ a~' the }
~.cab~.l ~"
(~) ~h a ~p~
.
~,t1 the case bea.n~ analyzed t e z
~,e~, us Hate that... ~ rs 'ram (6) and (3~)~
~~ eome?~r~.ca~. aptic;s~ as ~a~,la~
ap~prox~.mal;7.on o~
servance of the inequa~-~t~;
is equ~.vaJ.ent to ab I
~~
~ ~1~~ ~ ~
~ a> ~
~ ,~ ~r s .n ro~t~ cas ~ ?~^ l~
and
hex a `^ ~.0"~ s ~ 2~ ~ ~~
ror expo-e s ~ . ~ ~ 1,0~ ~ and the candy"
6 denote that
.di~~.ons o:C (3~) and ~~ )
~~he car ro~.mat~.on
~ xn th9.s ~,n~-~1ce the app
. deno~s that ~~~ 7.0 '
t~.on a~ (~~) rion ~rhere ~eometr~,cal
en se~.~cted a,s ~ralid up to the red
on bets
c~~ has be w ~ ~ ~
h~
,,
,
w ~. the ~,~.{,u,a n
. ~'ar small value. o~
z
~,; l~,cable ?
~,
opt,~.cs became. app ... ~ ~ anal. (1~3) ;if
~,
canda.~~.ona, o~ ~3 }
eac' ~n~ ~'` 1 f lp~ the ~,
"~ ~ when s~.n
worse but even ue~ o~Y oC as sma~.l
o lp3 , yr~ t~1 val.
lp~~ :and ~~~ 2
axe as ~o~.~.aw's: x~~
7~1 ~~ti at. ~ I ~ i t t 1 ~' ~ . r ~ r + ~ ~ i pS71 ~:r ~'iE i~ +'~ ; i y i`I ~~^`I uv~q r47 +i' H~~,tir~`yi tiu '"a`6..im,14~Fr+~ .G~A~ 1'~Ir~kl~~rnV ".?h rlGB.~ vibgi ii4'S
~(~~~k R4'I~~ H+ll ~p~ ~4 i ~Fl~ ~lou~. ~~~4 ~ ~~
~ E '
,fib ~ ~~~ ~~ ~ ~~~ ~~+~;~ E ~~r~ n ,J , ,".~'f~'~~`,~'+~~?' ~ ~~ 1~ Declassified in Part Sanitized Copy Ap
.. i r.~e,9i,ww4 ,n ~~~ .. ,,..~. ,r~L ~ !~ '?~, ~>>Y fah ~,VJa?r~1., t~
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end ~rr~ 1, :Cor
and w~. t
and, thus,
tho wQe
1- 2 /16) (e2 /kT) Niv in ( 2k'
VFW>~'~ ~i~ k`
ax'jtb etjc , VelociW
lu, U.
4
2N1/3)
=32/3"
We see ' th C' - Cthe anisotropy off' ~ is rather eonw
~,hat ~- ' 'tis nocossa:r~,~ 'to use di:C'feront va'l~ues
~
siderable, and that in (' ~)
F .
L
LK
?
of \ for , and Eoz
lose
1~en ()
S
is T~ less Considerable;
C
j
general:ly ~. t,has no special practical significance, since fUl^ ~~~~ ~
,~q ~? ? , and what is most important, it is scarcely
J to determine simultaneously' by experimental means for
possible tr
the ame region of ..the ionosphere for different components . ~s ~'~'
's concerned, th
as the absolute value of \~ a,
e presence of the ,,i duo to lack of suffxc~.en fly
r 3 not I?_4
precise knaw:Ledge of to parameters a, Ni
e experiments?
and T in (71) '"
? while for the isotropic case and for ms" y i
(23t/3)(
kT )2N 'ln (2kT/e2N11/3 )
26 w
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~
r`
c? ,
Ncollisions o:E' 1Ons.
to 0k4 this effect
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All off' the expo u obtained in both the iSQtropic case
and in the abSeilce oi' a Field are based on rte as uitptiQn that in
~~'k?Ir~'~'e
the ionosphere the ~iQ1d E is equal to the aVerag~ macraM
scopic field
(72)
S'Lion as to the validity aC this ?q,-ality has provoked rather
The que
widespread da.scu$sion and has already' been examined by us in detail
a ~ayrc~'~~~ ? ~
(10) In the course of was ~~ that we can
ion thi.s;,
have comple to confidence in equality ( 72) only upon observance of
the condition;
e N/ G~? ( t~
?e the op portufity to note here the possibility
We wish to i,a1.~
of condition (73) with one incomparably weaker. Con?
replacing. the dc. tion (73) is obi,a?ned
in (ld) as the result of the requirement of
smallness a of the first approximation by comparison with the zero ap
(lC)
for symbols)
pro:dmation, which gives see
~e~ e i W t~m ~ ~
b w
4) 0
~x~)~
Q In
re mo + e(r) = ?(e) ( /c,
why L! it
(lb) the conclusian was drawn from this that it is necessary' to rew~
1 t
the inequality .(U i j 1 ~
quire observance of
the ineq ualiti wh also reduces to (73)? However, the character
x
i s of variation of ~u t is just :Like that
~~ C
~,ic frequency
'
an the order of v/, where v and r are the average values of
i,c.,
the rrr + 3 - distance between the electron and the ions
tacit and
Y
ve
) X2.~ ti r
Q 0;
C the ~:.s no a ft~,nction of time
: wh.ic;l
n rv V
See (10)) and from the condition r
1. ref as e ~, ` r d 1 ~,,
s he ~~ Y~
G c L we derive the ine quali'ty
2?.
ified in Part - Sanitized Co
 Declassified in Part - Sanitized Copy Approved for Release 2012/05/08 : CIA-RDP82-00039R000200100034-2
U/f:
h)
where N, concentration or ~l~e ~aree~nce a~' ~ ma:?tic
is the conch
1
anything g here. In order of magni tudR r t')
~a,e~.d does nod , crane
1
and ' (kT" /m)aa and cgnsequent1y for T "s' 300 de reee KelVLf,
v ~
"r'J i07, and the condition (7Li.) assumes the Corm
v
e2 kT N 1./3 1o"
,1/3 1,
i s aiwaf S fulfilled under the conditions of the
inequality uinequality (7) ionosphere whence is yielded complete observance of equality (72),
Received
Ph ical Institute imeni P W N. Lebedev
Y
of the Academy of Sciences of the U~3SR 26 September 191~7
Ri.bljo rayh,~..
V. L. Ginzburg, Journ. of Phys,, 7, 289, 19L3.
2, v L. &inzburg, Usp. fiz, nauk, 28, 15~, 1916,
13 Ya. L. Ai'pert and V. L. Ginzburg, 1zv, AN SSSR,
?
L.2, 19W4?
R Hartree Proc.` Roy. Soo., 131, L~28, 1931
14r D.
ser, Liz,, 8,
181 19L)...
V L. Ginzburg, Zhurn. tekhn, fiz. , 1, ,
5,
6, I'1 . 1). Dula ovs 57.b, F. , 6, No 1, 191.
7 . i~ay Soc?, L , 2 , 1933; . N. Ices Senikh
M. r~aylar , I' roc ,
SSR, 22 r
DAN
8? Ia. L. A1tperts DUN SSSR, 53, 111, 1916; , 2, 1917. 1
9? V. L. a C'n"burgs Jourri. of P1~ysr, 8, 23, i9W-~?
a. 10. V. L. U~ nz zv? AN SSSR Sere fz., 8, 76, 19W4.
~.b~~r~, ~
28 -
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