Temo "Po' Professor E~,,, D,....., ]IF? Ir:K. ll;~ ] Cl" Re xoliati -l Lbrto ry.~~,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~J is~ Cspy ^at o and d a a —C o A.... -. \.....:'r y' i ~,d~ ~ ~ ~ k,"~~'~ \,'~~' "".. ~~~~" ~~^ ~~ ~- ^ - yy.'~~~~~~~~~~~~~~~~~~f 5 9`1 ed-ui-~:~~,.. KMS ~ ~ ~i ~ ~ a~'~~:: Lr:slst IX 169:~a~~. ~B ~

AF{IJTB5... 1.TN-.....,,' 1ASTTXA DouAeerm!t Note Al....60790 sT.'iOES }M B.ADARB C}, SS SqE0] T.,S NS',' Foek T}lae ox-*'Te'3h I~ve'Psi'{~yr of M.'higeSm, Repo. Neo. 2591-4,8-......T ctienttfle Report No,, 3 Air Po:'e Co:m'.:',..et t AF.9{6.' 9.: July 1958 AT rovedlA_< Tife:teseusae h repox:ted i:n (,s.c!.,,eimssa; u: sbe: b......... Air Forc............Ca B Ce.-nlBter kl, B-&Ia )evInetlnp,81lt ComIxmainut

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T HE i: U N I I T Y OF MIC HI G H ~ A N TABL}~E OF C.TTENTiiST rTm'a r VI a Twe;Xly~ me:$ %n:$isfX~ li[ Pr^l emis, 26' r X'A Exae. l'e $ 38 IT,:'Lt: {,'Je,.l$::$48 penc:{.:: I: Te C{lxnto:~:;/Lt,:_ of Ce::;:[n oslue: a of Lege "e n 49 cE, eeto an Ilitliy Many.eaed R.ien, I@lintaivatkntXf:I, i.; ^y~~~~~~u ^u~~~~~~~~~u^ ^^~~~~~~~4

i iH E V jB R S UE STY ~ OF I Ci GA N Z591 3. T.'~3 ~.-.. "FIriB pepri is tiity~six in a sa,'es griyniig ( t of StU of r;adari crOBS seetrs.t a S il TXe ii esi.ty oefi l4ss.g a sa Psea sh;i.si-t u e, T i oe pti:iOy xai s or x t is pur.ograX ar1e,-; To show ti t rai " i POs see. ons can be dee, - m. tined anlytaicallyi 2. A, To oblstoi!es tmeatnst FTt - toe'purtsing thse tadliati"Ton pt'teott erts fo ae t'eTai by a.pproxm ea "te te"iquec wati de ter. e i te paoees ti.to th:e accuracy e, quislred lii militi y probleT b' ut w'hieh do not %e(lit th'i e eixa cet soitloios:. T o e obt in mean4 s? 0 o ee03al.oL ~n g te'daotr cross Oasoe" ito of: ~ ~ario us 4eeets of I' mlit'ia, y it.eres8otS {(iSt@ c 2A oT rd aT e itno ort To"o ^ oy lko 4k c >ip2rocity'1to1 ore it 1is rectotorf y to solve nly n ee pf 1.f te' SC lroblwS,t 3C. To dcemeoinsta tte toa' tl t lese tetexfor:.tico. ros set ti 114 a.di toeo eotcxle~ly dcietseiid diu;a.lon pat I ee s a aWe irn agreeme nt wtlhi e ixpe t riP e t y de4:e~rmine d ones o 111 r dia 4t.e t $ vees arie-,t t o A,, To eo.mpute,Je ex~..,c UaiJeoret oal], o.es. se;lo ncs ket of var;tnr;s si.'e Ib o,<lie by Fshtxti-loe oof tte a4.:tOprSiacte b0o"4,xItv1!0a:'.s..val.e pobles 0nal J:'~em e" e le:rotmaiogr:'It. I~eey e..'C4ox'{e'1e 1U oexat ita-tl;es froem: nI f,"oesi-Irm.:.eo...ta ees 0 t/he su,;[ta eecei of smple shaIpes by tI.e so:d0io'n of appo:rop0 iai~'lt bJ,.1d&1a'y' vae problems a,: n.4w:from e~lee; cttmo awee, It:i30eo::i" (Sixnwe J[A F 1~B are interrelate~~^'<^d ^"C5.ae.?.>^.<?.,x&^.l%.-~ $ ec s i0 solve o:d' 0.y one ot tese lPtoble a. ) o eT mo xai.0 e 44 /..ps a~Xt0oxima 3tio tons pess ble it:1,p as problem an"4 xd to deottmione the li'~ml:~ts o' t. e, v'alig, 1tt, anid St4l: ithry:.}4 ~ To:inKd 1means of eo.titfn shce i3ple 0bo04 y soii'tiota ini o der to de:te. ire A-o,':''s sec t i )40ons o f eo 7s4 b 4 14ies, X~ll^ Ti'Pf^^lYI^ ^l~tTT'V~rtT^ T^^T'0

THE UNI 4VE R SI T Y OF MI C HI l GAN 2591..-3T.. t 4. To ttstobuudai;,a (Atusi it Zr4xt; t"[,,, }~ fSutef,;Se oraputt ts 1to be dcnixe qykiy for ar bitrr oets 5.o Tor 4, hil taO EW4 iU O k Ace 1. fel v~~~~~~iUi4

TArg: 9 5Jy>4 Xyv73~~1 y E g r} F THE UNIVERSITY OF M HIGA........p.....1..a... B rbef l, we lelieS. ro achas ther ]i theor It ts a mtcctl-kt of l& t.intca g 1ti ~ed bydced l- anj ~s %:: -:. ee.ts, -a.tc w.ss re. or ln ixe to >'..rt..e of e a g cm wbictw.. V. ~rStRf.r @is trited tPco be i mo0 Ca Isatra'~ c' a of cta r-Ist-Jicc denotukacnoe $@3 ~ ag~. VtkreYCt~ rI Sto lit";' te ~ of lie 4 ~Pont rac aon. Te

T 4 1, UU 4 X U B 0 M -^.{& o> MM X C. <$' 17 "~?' ~ &' 1,{~ _ ~..... t E:..>.'.:. 1.:, a, o.,ts li:... x..t. e ofk lie ofTereIr~,A a,,t dVtA bry a perf~ct~y cnWt glt jnx n i ci Ketar eV4 Ynder' For t 4 ras we propose to' rev the of 11-1Y.44 V t' 4 4.4 Vylin er $tAb P We V tO V fr somw lK. t to h th e wv ~~~~~~of W,~~~ Fr (111 f.<r@| 1). We coaid 0:7eC< 4.,. 444'2V4A irua w2'<,4 I 9r of as li' I. I ~ ~ F ~Ifr,1

u v$,.'!. Y ~ O F N:,i C HIG 4 AN We ow sh to e e the $ag e t$ fe Idne a thie ui fee of.the e ude. If the ^ i e 5g ~4, t A t) NN 4 g&rtkick 1l to the tie<o9Lie lali &e r no'vani14g 4o3poneut o if I Ated 93 4 su5^ee to In te Z9teot 1n e, we P.. f acali i 8 is la "', 8r "r P sito ae t,.,,, 0.., (3) ~- 0 ci.(j e. (1. 7'............ I 4 i j 3'9 1,/~~~~~~~~~~~. ^9 $9 9' 9.,,,. 4 24 &>49).

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TH 2ES <E U * EO. J S UITY OF M C? H CM AN:tegi and l e yr t 4tu y e (eA. 4) et Jtuo. ef ftie nteg iwe fid S< ee I.'S*'..4.....aa eee 4e esa te (e 3) he a(. 15) 4;oa{t'd} {ka}.?..'~* "s.... w azl %o t at ( 9 t) o s (X 11 - t 3 11 s 31fg iak'#4) e~ ffr> xtsaC{ t )4 2aa f 4K,:: o....... t:) *to> (".' t () }.E t,. 1~'-':dt- (i. 9 Be I of ^~~~~~~~~~~~~~~~~~{ 9 $'{:}~~ *** -~>."':,i~ t................................ ~ ~ ~';' 1) w'^-S^W~~~~.~~~{.:~.~ t}. ^,,

wefm atrie t ntear ze x or ps.... >;.. >. b.e. e as v in i.^~ —<^ 5 c ^ ). Aedl ~.....: V'E:~.'1, an4d,we a-vu e e' th ontor &to enirlte se e w. v h Hes....ein tn fit qadrte 0 0)tg F 0 ever te resiIe feA es erVets &~tmy r 0 and fore-x'!r a xfS ow < 0C9 )Y,C^rketed be a S tus..> 4...... 4 Fi., Xre 1 2 Th 5otu V.L/

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Is'"::':.M.B We,, ~;.9 ~,ti, We n 8 d t A ~ o gSae an evaluaed.a reside seres feor t For c regin the Welon aa- - be lae ctieX i V 29~r ax] dr We e hMCY&ir at4at4 lie~ ~~ f ~ 7%e ew ai t tie We m Pocwk's W i E ef. 8) w kby mean~s of a si:'mj% at..at g(2 9 a. n terms of a He do M euf atio e invot of Ws I e of Fi Is c le >et 4 (XY )A 0 e ei4i4 w' e i oof a* yndc1 4~4) r2( ce44 &.%w4 424( e 4 syst m a 474JV'09414t4 44 >) 9 ) a >4~ 9 w <> 94 $49 4, 4<> 441 At tX;X s~~~~~~~w Se~~~~vw~~~ut~~~n Hw,, (t~~~~~~~..'4,

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TH ME B UNI VE R SI T S M ICH IGA ti mi li ~~~ 25e 9ofI * 2 ~ It d PL Y t. 3 j' thi C) i. WeAr V &Ala.{$_ ^"~~~~~~~~~~~~~~~~~~~~~e,....... ~ 1 lie44 T~^w (e^ (1.81) /':~~~`'r(~;~A ts'ate t77 $ 2 1)4 ve,. 1 ^ w, ^ i:'Z $('"12..it, 74~~~~~~~~~~~~~~~~~~. (."9~

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THP JHVW STY (FC M I V H GAN o4; 94$ *6..'...e e' We lief h tn t ordY srv 9 wrtlw trc stt e 59) ao 9 E WHle>& ** 0 (1z. 6,) 66*6* 60IQO If we""to:! * r..S, f.6..........{ *66.6~66~ * 1.1 3 ep e$ order ame: we A..e t a t.> 1 9 3dl r

&T k E S E ER XX'.' RY F M C G N IS fe l W{ I ~ n@ B. S I e, ai 1C! Il., I.:,, e' e9A AVF A21 ".....e'fi16 com, V}t} >t. on1 th ueI by "~..}t a, i:ti,g 1. 3., e....... "'. 45 s j (I, i + n U j) 0 5(1 1a <SVd^^.lla as a^ prtlol r s'l/: i t0 7 wWft-8 tW.', {it - Tim I' ayl w ~ r uaiIL'& a ^ a'n'i~^. t t. dtt t t'i 1^~ b, (.a

t~fr U9E, is.... 2...'':&t /i f i,~1,74) I I e C h ( F k a (I t e).......~... \ f ( z ) w T.nt 0 4we' rexit the we.9ng4rt?'9'-'$' ine r)p4 tict t and goth t s u t9gx< <e ts ( g f)2o d {A] 9<ln<<g s gl D:.XQ @Thsi':m.oeses e ep"ta. t.le we,a.:v e a wave.laxc.hed: at te bagr~ y h the ~.x ep. c. r~a t su. c.'.*e -*to." ollmbo fle...A.red.. eh*.a; on To:,t?.. F.e has H ewe~ ~'er, we 0.ee t.: b.eas. ca nom sex, e tm of as t ent used t, xPc k i bis d- %vp cx o e p

T H U VER OS M rY OMH of tM loIg tihe ur et e tof lee s & ere t VPJ JAC) a p.9e A of l tieok'th:I we o se V Ji k eq don aywhere I. the shhadw rekit providcd we defire a e set dof vail es and F r eah ireenvt we lmove 1, ie A,. ehadw T ti te wt A e w te fcuf tnt o7 { A)a'1B9Xa0 10hg1,|s~, cle, (?, 44' 44 s Qt V i I _ 4244s (AN ), d rThkin tesu an: eressicnn vto,,' fr 3. G1atiAre up,-. s'v wle) KS ej l W) I1~ F ^, (1.80) IT

T H' N IE VI f $ Y ~ M Co. T H }I GA N Tib^ r. l'Tl ril'^g AAIik wont, n afi ifAbJ B. h a X e L r ft t9), E Her %xttxL e V..... a It a) hxticc a*? o % cr a Ab siM. a tA Abre We now n a oe-p l — te ]oc rimi et tl iz;m-r bderie;.th e I.:}''e of;s a y 1.' 1A0)'^y l', we ale t.- se.s. as 1I 2We I / la ili eeI it'1w ra A & dors, @ 11 1,0. e 0 1? 4+ d t & i d ^^^^1^ ^1^PI-^ a' I ^l~li^ ^^^^^ ~t IT'^~lfOTI "Iv~^^ ^ ^ ^~ ~ 1 ^ll. i<L sa) (49444 4 T)4> 5~ 44 It fi \4 if I )4 j 45444>

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.. l as'In l................8... eI I / / \ g WSW t9 be inclde,alcig',) ex.lnor,,aus,, cg te ellipse wi9=XjjM....................st0.. tw#,2. h... grnicadc,;e -1'e e ]@>S b y.g< )gS.... F c k k~~~s ffltuncra,-,,'< cxr' \ "~~~n lk"~ 4 ~ it So~-ec+ 4 t \ o S 1 tir a~s(e)-~z' f a 6 (.8\ ) / I' 0 es{:.a1:} o a, ~: )-B(%}v....:)1 t }.{:,,s

H i c Iy N i VB {i~ O S }I [ Y O F i CH I G A ai { an:d F are cl tll: cr kk t he S>f s.Is r&:Pe. l ~ll.ipi. f _p) -i - c <., C I- ut'i X d d!4s90 4. w:.'&:....~- — s e ui~t~;o w t I kwm A O? 473444 W, We lieo, t ng vv,,~,ive o -!,Alii x —-bs^a~l, } C ig w@'4tt>. t,} tti;^,.2 4'6,3.:,} $.8ee ~,; w.'3 8:"' *... {8..ez8 / 4Z3~47y A1,' > 48$ "83 gef og"t*} tWEio>%,6o7S tS jC.Sl.-} M884888

N IP4VERS ITY OF kICHIGAN 214, ~~..<. These aa vug(bts- g:,~ -for tlie sdow rgion g"~ es g() }+ e. g)(: (I 9) I'. q ro..,3.,:l.. o IZ) ge s 8a a 8lie c hv CN):PY re c~r~pt ~wic~u v~t V 2X1 ~{23^ L 21 iwe ft.t -t t. 1nw t, 2. Sr~ j 2 02 k lSW -' ki 3 l ~ 0 + k a1 toI the to Fas li ente i~ Vt' 33 %t~~~~~~ka la + > z3 0 Vi9256Z %? i tY 2't 2 0 (1 97) Again3le to lta u to 4bt by J3 JL n 3 0, ~ ~tw~otdp; 3to212333%t21C lb~w~t2 -'it3 a% %3 322133% lie In tli~~~~~~~~~~~~~~~~3''

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%HB S < " Y O (I.M H I N ) cV 0 y, k;+ kf...1..0. 2 ene.... ti. b:,'J is om.-o sded yI' A1 - 0 pA lded 2.~~" rl.> mA AN'-< $KAA p ice. p a &"~''ej:ft ~'"'' e.;:"' so~z &A 43,,,%, An and &i of and be bce y i we Sa b~tie.- LCAs4IY(1kA 1asry)_ AwtAt.0 ~ny.: A (1x 06) A- ~1 CA. t ~ - n-. 1443)^

lT9i1 1U3N w dtr tixn- t Jlx itse1f x0 e x ju z atio r fixe- I hk r AeeI x ou t ItB d8 iap8M M:X.ige - e at(B mef g at fa sto ok es- n yg tA vSay tp~st4a> I sea'stx

T~l HE aUNI ER OF MIG We -now,, rn to the ft to Wewst-r -ehtDr.acny yey -e. a we v-str v aer A~ a the ofi l ie Si rtul we. 41 de.t n Vstpi c aA t.tt lime d ttgo if e e of oat d QB, IQB of i (2.a W) 1at idat' a aj.wtt <aag e pBOor a n toY ti a thae at>l' the tut tWI'~ c'1 9an at'tot ttni t 4rut by tt t''& QWnN'' a 8 as ^ " ^U fti^ ( )-5 Italy) ~ ~ ~ / t r~ ~ ~ ~ ~~~~~S

? - 00 ( t1 G H wB N th we{C i{ AN WNsoe lohe t.I Bit g'.:Is a..lta e t',tOfmB of e iitn e liet oto oa:d ftou %: and havig simple poles {(Le zeroe of ( (k) In ie irat laadrli we hage the / @ x <>..I cow at~on: C to C 1@ 3) P > 0 R-nig el.te)' ie le c2o.egenr't ew.a~ios... a=O Now v g use of, e lt edt j ie ofnr6ncnA,-i ) we h)w av ~. ~ ~~.Rita 11431 T ( 0s a e 0 n- P (4 )n+P) G:)r ( ( 2(

NHIE UNIVERSITY OF MI:' GAN R........... j (. ( d - J; (0 XS~g) >t;{a9 4444 -m s42' j e > fcf > - ~ ] s / t4 n'.., 1; Ya 1; ((, 14. / 4<..... w.i.?~,~';~l'"'],.. 9'. 4 4).4 4.4.';< { O+g,, he Oa.,,~ Oa-I 5' 4 44" 14 4.i ) ^ ^^ l~ ^"~~~~~~~~~'~"~ 9~~ 4)~~"5 ^ ~^'' 1

tt!I'L U N OF V 2 l $M C Gr v ~ k' I I A N 4.tI'1 (4:' /'Mip'. et. s:Pzt,~:"?*- i: A _.., &...,a u'* J,/~~~,t3t,<j~~..~gi t' i I146"':fl t of 0 li er iee~so Iof lieloe welloif noIteiW^^ gf, ltxc Ivaf 18f.i' wv' jus9tth Sp r lstw es mes d f x ol t rles 22R~ll; tfw.&&>'22aS a&! ^~:^^^^ ^ifi S^&.~.^l A^..<;. W~?~^ ^l^^lHw II i^ l~lfiiili^I^ ^1 iih iiiwiTil T'^^tlFig4ur'ew 2.ll' 2Tt^l ^f ^ eflw elw': hhky& e,&~ d.a i;~^ fM&j unde the WbWt oftl>lfe^ 4?lW I!l>~^i^<> 1,383 (c^fi(013001 fol~o i metit l - i bi ao l~ m o ~ tc o te(cria, o itqZ ni

H E UN 9VERSITY OF MIC CHIGAe N We S ( 15,) i I F ge leof a sBO ti th A I a a v thn gt ai AS ad it <LI <a m 3 ) (Z. 19) tLf, Is of.,...: acd hataCx1g Ji lie ia:a t. U'' 3 ~ -^'' a^' 9 ^ i ae 34' a 4) 44ad)..B...a.orig t i dy ato t o: 0. 4% We no it &t 8tg..S> to (th2,20) Ist<:' t sr w fi, ike 1 tf>so ot l4.:-a- In (a11> Fa ( t "I +It". ea ) J^ ^^ii............. 4; vg ~ 44 41 a P a"3

Ir' zw N h' S IVt 1R s OF M H t} t I G ItAN n In dia ~e el do)% region, he evii, tes ZO) as a residue series. ffiri the tro pole ocac'rs -ar PT js s -S fc aTfl'Uj I ss> B I ffie i^ t'l a > tJoox b t L'l fi xredicle seze e,,1 zavsag- ('i t) hy "" li Wt iemai it p- rpie c oe of Atp B k tavwf g h (Z 1) we see at re is u or -,~a I~'dffere Ie aen thze Br e esxits c.d tihe pi o ic e tate re)ux The pty& Ptl tbignlft e on f a fb en Nis flitfl. A e.' te zL'... " " (2 t,,,: /" iP ritcll iTt]e:~4eit 1~s tp it it.. tmmeta to oe nAoutW t tit it r Ua Kie ioe ~t. w tIp aa.g &t to~ a twyo.a mei; I~' prolim (tit ifnitte okixtc&'ttW )eG y'Iile;siJtte (' bez a Mto1 C P..a I. t 7 ves e:r'ee.: Int tie s'aw oeAver e o n poe.peb 8 I,, ao.iun:.i ti,"., I. t. in Elt (> iT14). vtlii C rIf we taoxide r te ir ite dets t aat J. a'e gSir1t 0 ) ilit (2~~14)~> togfi ai) 8 ~x-"',mult alri 1 c noted by Franz 994 f 1) 0regTbi t 0 601(t) I anM-l tttat <P t~ <> rc r ng j\ 4&tkh writt a ~'...:..... pd {',''eA?; 13) B - IIItt::~,(, };~.e{}.,S~:~ ~x:~::a:e'{.:x""4P~,d.~""i:, e~~..~~'..{;,;.s.te?,''~-...'.~s~:~'eA~~../i~;P~~,..:::{( I; -" —f~l~'~IO f ~s-'.^ 8rl fpt'^ J.$r~ f~b.^ s)e;.}~'q..4,P~!:~.~ xlw::~tTe;l s;t~l.e,T.W....'l~

r: iH t N V II MI H I GAN r i 8 I ^-~ dt (0) k ( W( ) I e I,,4, 3.. 1 Cs the le o Ne lsa pole 0t w w s i e use -1,21.8 asgy ptA-ae, rpsgelpsn', itsenftim,% 3o ( (0 {2 (..ta44eTt 41...-n ai: ke a d dt4eoroz i to' 4n3t 1 > P;)( o3M4ioI oh are A's 4r 4l~ 9r: t".at s sa~ i9ng 4 tie toL a, 1 L *. 4.. the of4. " 1>..' 33....the "'"tm rfor la 20 the,3F g4 Sli""' 3a.}r I. "" beti,' ea tp a; 3 -'4;4'4 -e t'4A>i;3~}.atile:PeaS. o: et eS., ~l/stt %gi 8'""'l""''dt}c>.y r >~t> *:' 4a"-,. n.' ""'"" ~t.-'."4' "'"'"'" sR i.s... t e s Pock Q(Re, 8) {l"P.f.fe"d I ihast maped itdo th"e p:e.e.. ii Ipo',, e saul Is ~pla~e elecrom.oetim wa Wve, We ee th..e plate wave to be txh..eloie a4.4g it.e,, x-'11e ande,.':',:/red eret- a acol flnalt,3 te t onxe pot the s.h.a.:, bc.1ary the.,'i x.orn. 1,to 3the4 sur c an t..e z. si phose.e e so as s:'to ae.;,.,}:.;. ystem a~{ aas is 4e 2 3 33

e As brwe tbfav, wt r:ert aE:ar t ri We t" E el -.t'7 xrd3t d;5 44X 3 (2A27) fro3un, f tA-tma Ut cnya 1.l' e. dkVn:f1 r )P @v t.ee. C i 9 Nowm we texuxiad dve order arxgumevrn axt, t l e1x~W 3..U ~ fC T 3t cognxripoine nt4a fi xJU 00 1I.J.Iu. 1 W2 ~~~~~~4414~~~~~~~~~~~~~^~a C

IT UNIsV'ERS tY ~OF {M CHIGAN 259 1 ~3 < k ay ox lii a 21% -i = e {X? }. by tieof ( 2. t 31 we M >> m.t>> 1. (2,09) s i0 a.-ad in restrict, we;t M m~ 30) gog k k 0 (g231) x V ]by applyl;olg |ig e:..a~d(ert iF assu"spea o E)qy ati {, 3 we ha re ti tq:~a C,,., i / aR-' _ a% aa -ult t o.E a / k ~ y r a-Z ~

TH E TI U M' }:i~ R $ { T Y OF M C H G N %' t 2591 0 3 -T We see then tat ti thl &eetinsioal Aouithwii o Ptck a~re pre.>sely Ue 0wCmeu t oltuitt s gdve abe... we e g z we t' tite + z:d: * { Z{;} axid8 yt"* 8xa CC 3.e (2 35 a'8 TI ce $. pmSeosety te,tl lo te,,.!% th 1w d,'xia pt Eae g..e oave l, te irt01 ~:.4il.:}.:-ae y b, x a} we 0~i (i37) n 0 p s,hsea 0i, 1~Int m —cmesiw IPr,) —ilm s ex s.35ei

Thee arWe te ont ge In E f W,YeA Agin x~estkt..e a cm thee teon d X geometi, (I., 9bw 11.1 ta Inx fiex e a ok)o Sie. tiYXg us~ e of)i the&;i dith vef g POt>tiite l tiw of cc li 1) Soc do lirsuigh.ke wk li diyen U i' g& e of lie weO ( 4%wr on h e e Qatv. rM -v n::'0 rAt.l het tihe"b erx arc SI Itco.l e tido fgat al210u o tesson goin t nto I oK cev. ItO Oe of e tX.flJAe 2 IAO tQOe Crleiv ( 1x th et.O 040 2 Tsecond neq it fae 4 aic&e of oth:tOOinmltbQ Ios Ce mJdir 1o z.. ~g g tei e A; PM a,0 i$ i 0a f o 3 6a eb41'11%1136109 (1) tliev 3a he113 9 9 at We foet;9r tl e tJ c (2E.3) S^Rd( " * 86 4 9 1

THE k N UNI VE R }S ITY (C ii ~ iM } C H I GA N.. 5 9 --...... oq, % 1m dZ 4 i! tc &d andr; use o ti tF andkbe t tmth, wse pqint ~ fiel fwtin be I tctor.'nYctipi1tod by tos fac 37^ ^^

edr " -- a' "'-"-~. {:<$OF M I.'iCHIGANP: AtN ES ~tXY.A.&.s'44 K 9 TIM CtP. As'X e}[ll, t; co 1 a ea o Ms' o f, J avy r or wt Ue ti pda.' t dt;?nrtii 1 e (J e on tieQs\ki of a cytx ttet sw -yta'4Oetly4 a We tan i:tJ< e e(leh Lt dMkavLna-i WeaL:4R st o fseti..... tha t trallt of tip, be bwe:1. t,, an wt.:f.'4e $ uJ4 te I l t o... H ewe atf1s1.2tdt an aipr!im mthod of d e t e rm ^ *ng the fil e o a ia ewe lapri cul a w it i the i i ctQWn o' de to beuc t nt a tll $of tie t 4B lbrtu h tjd ati4 IIt k f11ieI tliepa rt if an Res J1 sh.AI nur Wefltk it Pr t ~4q tte t4t h Is tt ft 4 F an gn Rt4zed ater anI.dte of<. Heil tr.o h E( We -EItket te S d rettrkm i 1 Par <et be u i at. f t U Cw e is s34 o ed a1 appl the ra344'FW te litli w41 t dte i te tiA onfiti itrue mnvtce of th con e e In trh;tn 911 icat jran a,i 4ent for fth 32U4ee alea, Pan o.Vf& 2iIc {ry'D I t ling e coordimte sysi m'I 11 stSr, tihetd In Mgu, s~3. 1 liea~t,3r ecwtei t a oof l, i e c, can >be2 a gi w o t ~ j X e2eC~<~@t ~~3tg3 ~ <''l4 " G i( > i -S t M iS >e.g~~s"'S,.<.'S~f'3

I tol Sy.e -- Co I e for. -ie. i...,?."..

Ta R $ T I O H Y F GAN,, ^ > (3 -191) ze r wme iddn.t It e'xl plae an a aJe 1(1 > - ) wia the e tis is 1%...J, n 0.a $~<. Ut I;... ti)<g<d9g by:10 >:xt.q t.t....... t; 0 (3. 3) whle4re f4 0 is the eahn of. the c* 1 e. in Ca ta 4-te 9tC " 19.te Lenotng an 8 t 9 the ee, Ioi Je La- cw buhwdc b tes we' 4ve L tie sohtho. o (3. 3) Cos -. co 1'.5) taz 0 Kt' i t.at t ie ooof: we 4 ide Lt- ield in Ito siftadow as wret tg r a mrave aItntih. at ~tthe. hIrd. hotndary and prst pa tIi atl3 a a3 ew wesic a od.l.tr ng to a - ~ ~' (3. 6 ) wh ers slie o is t of t w W nd3e lioi.s n vaturavidre la e wthe 3a -eS Xktytt tet9 a( a ItB a to liere f 0 p to 1Ke lied dt'. g o ufom td t i~t4 f the geenitolbetI /940

HE UNIVE R$ITY OF MICHIGAN.'.'........... We need teia, o the aQt7h.o a'k eo,,,'i,s:.'".......a sW':...... m t3 xse~ oAndow P.. 1b. ta a at.i s, P haIs t nt A et 3 $e r T k.. Thb e4t......3... d r, ar a e 9*9te9 ^ - o (30 7.) lie a S.9)C.. wthere a Is to bte'r Y <4 e <4,, et.....w...:-'s.i} s ( I" (3,1 39 a h a 1 [ 14)0 dt tw&1 36w 41 99 r ~~ ~99 dima 0 d9 0 10

~591 -""" $r Et B J R~s gX~t B RS E r \ ~% s>+.-:s I... g.:a o + qJItt. eto _ (3 9) wve; nPa % " vh~e I~tp.d. iao 5e.r. )xL Ut>yeAE ruCEA&LE~d~ttk r' i.udt ~ ii r' ky(ea Ad dar tdat T~eb t of lie th olts t,'W,e Is al 14 to. be (3414) T arm., IK sV>$, O r (3 153) 0~~~~~~~<a

t 1>%Ett F X~' g ~ t wk Xta~gtssXt gstm~g~e Atit.s @ i. I 42 I i~llm, byffit: q1&43tt' t, &M t S,,I: X2s9g a~ Ci AI a T ( A A 4 6 a t 1, -u, I i t xx 0 ma a t Ue tlie ea 0o', 40....... ~,~.os d44 aa'e, t:o *T i~~*'":~':~..... -7 4 4(4' ) I w 4e4 ( 43) b>. $5Z& SX ~ 4z 3''4333 3> 442433 13 t 244(I 4v4 ~ 434333 433 ~ 4 430tP~4 4444 4 1I ^^~ ~ ~ ~~~"4'3 24 1444 (416.t341 (343) 4~4433tL~%4(36 ^ ^T" ^^^ i ^ "^ ^ ^"^~~~~~~~~~~~~~~~4 42^

Ie0t AB te nMw I ^.. t.. V' T'.e.....'. (, > fR eO g^ &'' > X g ( f g)4 455.~ ~...:...I. ~'l,' I JieF({,., fft eFe I.s.g -,1e.M.'' I14l' u tel. perpendicul.ar t.o the geodesic. given by..j H~si~w~S > (3...s.5! We i ed the pwgje oEdn of~ the!magnetc polarization, gp onto the per yef and tangent d4r5. o4s at thie.a4% In 4 order to app lythe above te of 35 4 It Is&.'3 ii mA 3 9)1 6166 ^^^y(i^

i?24 It U N t'IV Fi I vt Y: M C H t A N Let te poi.nt In.ete l ibe spe..Itre p eree... t.e.sr. CO. 3 r l os ee se (..p'-*.1 a' 3 (132) se ae<44tobesv<ed4r <4 <4 44 ~4<J<44r <we note<the4~rst<set< te44te a *......'in~~ i 4'9 < A 4< sin 4 4(3.33) co- e ~.,...... ~ ~ 4<~4' >` 43 aXz~ ltmtio. In generals t^&4 he a:...........ao..b....a'l: 9. i1X *.J, e'a a3'111,31 etm $ h est.3n a gi a 3 3$ of*'"3 ~*...."' a....... >ziJg4334 in> > 4336 E;2f 3a1 [ (guS S f 4 ~4 34a $ 4 44444<44<444) B L......0 k; si., 1f) >4< ^&3fra4 ~S^;^ 4~^^?^.'24.4)3 ~ < 4? g &~23~K^~^~> 4 < ~~4 44.<<.. <.&:t4 th

P H tJ N }'" V R $ OF } i } MICHIGAN nce we will. ead to o t'V *f"Ae Y PO sIkl vee t r a. P we hoeose Lt tdllttet t:'s by'resol.:ea "fie o.:mi7fe3ie tond ei{ """' su-te I ff&e Mf an r.re ion. Iw t, treated II, lies stI'g tbou-if- v* e bu, T i^~ r sf^. ~ ft3 TIlni ^^^^1 lk f tA l.... wale te: ieM *..ld lieg alo.I ie vector (3".3:'8,:.) ~.....;"'j e:,.~ ~t%1 > ggg i 9 t~ ~ a A ~a i~~~~~~~~~;;:~ ~ ~ ~,A * O1W xTo' i.UW LA y tvo awl t tOsi y4 coo l^f P4tti 4j O <X"hj;5) st 4 n 1 t{ l.. LI f'm'~t *3 "a p t: rP 9 (3 31s9o?ti>&<<g~Nc wi> i,^ p 4.(.t. <5% q,& p 48, VP; q~Q~ l f(., In~.'4"' (B9% 14" t j44 f ) n 4 - 4v 4 9 441 " 1;^. ^i." sin 0 4 i A ^ / a-^; fC ^^ f~ 1 t ^ ^ - "^'^ pf ^^i44 4

T ~I11; N: I 1 W UN1V ERS Y O F MI H IG AN C t U 10 N 11 311 t A.xto f lx te o g e Is':.,renty 1ree x...oi.. p-r" Je is, a o'J.~.:eg WS g.~ o1 Fockt 11 t -4 men0onal. W@ st-.t 1. 4w.rolltg t'tat i *t.,..a: t, r r, ".teg it at w J t:.e "4dob bt da:e.O as Flj.........2...........;:.........?? x \/ 9/ "%.,:..%j /':.,...... t$d'X{.'.... If th f tSI' If, I ry xtfit.Ca an, i4l c^ t t a,( cxulkaeb,'4i Ytm,1 le F, u@ co,. t twe e,' @l' < M f.ad th @ " "-.. lt Indeed o)1)4 @7V[[ th 1e )1 S@ete o t f Q doy du 4 Poc' m itQ>.,K LQQtjl 1a (ti P his been u to et Q". i Ae t}i... ". <of'Y an Q%... f'lt 47

%T Fi',;KT''.,iS S'4''' O' M C' H'''GAN "' 4 "" K)N ]" NS.,We li,,z,l raI:e.':e F k roacs:a4s ajLin from a':.....;...., 40c44tK.al lof tO& i e,....e.b...:. lPtoK)>"> e at li' 4/eo er....c,:d 4 K4..'b........ yllnd( 4 a-tit th e hr gh. a r et: off t t sle and tr s on rg. proudIdedwe Ct regidt j 1 SA sni ch I ere Is a f _ussn ef a " A{}.te:, o - use of the me.}c In ":l "g'~'l..... pro l":-'we........on It as O >244>4>4...g. >4>.. 44 <4,...........,:.KKK::'''4:.:'....442 "'Ks "':94' 6: 2QoF SQ.:: 3, ~?4 z>4: Z::,.'<.,?.~;..,K3! ving a asM. tie arm of oaf, i tic solut ry v a lems In which ffie bo ni a- is a lom e swx e ia At In whihl the Isemholtz ei tf e t3is to s, we li t the c f d of lhie 9- tio's S.aNS~wS Sf,3?(ssoia.l d3 theseseparable b a valuSef p em n b f more If we. lie t Fock solution is wht we are As f r. s oh whoe Id be c W ho'9v'? v.e sx'3A,s',9 e cf,'-, tBotL ofR Tht f.K " v 2) s f'' D< ess' ba:eemy e a n "e,, as f bot trofoia fat' f on s] e been "by 0m t r Forice Camb-) (d e sgkef a. S ent f r X a ethe re of N.Let n L e Thea fot P4411 ve K44~~7 o lie It'4 e 0>.4 9t.99 44t">p of 0.%2 01. g 1'4 on e 3 a x x teS d b y F c k (B a:e". 1 4) toif r,4-4.5 In steps of 0. 1 ~

T H E UN I'iE S OF M C I G AN *PA ^P^T PL El Tf' It I>:i, Q~iAT'?O.H'T{E.& SHt; F;I,} I'S 2Z Bff..A N,6;:;f.T{EETED.TWI1,1 1fWIIi C1E 33 tle la3 i > te A < I < 3 we, th 3 rpresenLai on,,,,,c,......................). { 3..( _ 3. (4 n,A. r.,.:'144~6 " Ai N (l I. -( l is"" +.:~, <: { 6t gwI:3......... s:: 3 i', e:................~...~...........4444:~<.'....',........... 4, <' 4. " 4'..,4 <4 ) U It~ I I 4 4,. 11: ) @ \(4K s...44 A Fe i. fi\f-. B / S'^ "B /f ) " 44 jL ^^..,s.4. 4 " 1 ~ 7 411 < y-~ j'siR4 I > l i! S.'I.. ~. V.^ I JJ 1.,I r......'. 9 4 ^aa.^,~ ^ W <l< I ^. ^ ^ ^ ( ~ 8$ )

XM Ut N I V~ E R tS I. Y 0 F.M I CN H VI GW A N 011 4 IA-~4 ( II) $, ( 19 (A. 2) tieo-bA3 I 2$r mIf( 1A I, o..... QF^ 3 Rs iI ) m) 41X ( > ) g 3 cV> + e + F - IF, e,::: Ws traio we are led to fdle l'e {:})D ct.s v^ i>) > A / a ( 2 w E ) frr-(f) iv /i (i. i 3~4 (cobs (4 p Oet'ved f(5) 8 to,'ae 7) 1' ~~'~'1. 7

[-1. 1..... we~P' ^~i e..... (A. 8) 4'~ th "t 4hA/&~ a pet g0) nm~~~~~~~~~~~~~Qe

t nIAE Mt I C HI G A& IN I We F.w, t N l 92 n> 70? 5 S; 19 2. M11YPn,Ci al eg - Report 2591n4, at -1of958). MM"::.'"",D to,,,,,, lie r by ri,, 4~ ~,, Gr, 11, 38 (1955) 4 44, 4L.)d.2 r4 144> Kt4 ll~ Ave&L. 5 B. L. V [ntDe.... A., (.'50,.. $f,;>c,, z@'99)9999 994 &TW Al.P Ysh1>99.S S. 1< 4???49<4994 B:A~ 9i94 9 " (/49t.?.3? >%e3' $44/$4/ 943' 1 91A4V2)'994494 6. W.i' <z 9'i419 ema9(AAaje 36 (I 952~ L. 2L.A lsn, T 94 aC s\of 1 Insi C:1< of Rad - 9~3A12899 49554/) ~...8@ *.. t.yee....:..AS.,.g.;.... <....... ~.,.;.:......... <...................... 9M) T', A' 394 Fo., k'A,..1 ~ J99 130O 4.AKAC 6)e.41A944 24 C4'.of A........ 165. R, AEg. Dollo,4 R.F>tsg9. I xodrs5 &..Ke n., a t,:. bes ~,}'iix.,,,,. M.: ff,-,i 1, "S xs..........d, e...t....?529...................................% F. l 3e? f< ):m V 9A.rra:y's.K...d',,'4:9..on.:0,,:..:A es"'.........tv Mot:.......A

PR A g CoFLASS -.aF,... ED....C T4 PARTA:3e F( A N CF P'P TltP S OE b B: t

PAAT A W t( S<25,^ D~ C. AF 29 ",9~p, ar.:o, Flor AF 518 t;etec A.i Unit",;ersiik/ L'I 3P; r Ap5.Coi-.u.de Air Foe;,?e Mi..sie Test; Cen2t.er PATiek Ai Foesre:s tse,, t ATTN,...... M 41 (M -S C. I I.D), AFP 8 -T.';,c,;al Air Group,,; AT?T'1'u'ajr R. L. Ste AT t W S..............,.g 6AF 4:3 tX Air, r AF 653., de /,,,;.d.~.... Airso:c?':":.e:Ai. e'.$..,s e.,. C'$i A 6'' Wr.it......Ai. "vt14..el entg C.enter& Wr', t-Pa L.t erse'.s Air IF. e aode, (' Y.a:: 4ATT

PA. RT A (co:Cndn.eVd) Air Tec, hrdca 1 ]:'.1 ee Ceniter'Wri:.e tson Air Fe wIW..x se, o". ATTNh A FC!}2 -49 Bhl, ACF 141 Qc 4m9n r< ATTN: WC'LT.- }. C. A., Trae', e' le AF 12t4'.se aY Bornei^ Air't TBel(OTmeI, t C wa or Griff is AiT e V:..e:, e AF 0 r; F3 e Air.< es entw kw, CsenterA Grl:if g":.tss Air For~e Base N. Ye), - r. AF~ B],, 3 9 (3 C"'r1 " R- C:t f. We rA A. r F'"orce' i },ssile.1.':%tAe"AAe SAT A'..ce e Hoo]emy 3s: LAir F Bie Ne 34 it A'Ae.e N'ewM e e c 3 AF 194 Grtf-fiss /ix.; F^ o. es't',e'~.a~ e, NPX' Ye S AFIfP5:<Arfa, Am:e: ATTN:- 2 RCOPP A l F 9 6i Comm,,,:;yi r cx derC WrigS.ts; JAir s A,'I4,et Ce.i:S r WArt:.,.. t s i Fcr'. ec't e te Base e <..'"4 "'tQ.A TT N Of I..CR..E \ tr. F.4. ret Air] earch,(, lu,5 AeAf'A and Wk s'Ckh, t...:-A',;5.'D,,<, C.A. T T N <,: CeL PA,,.'.,',.., A..RDZ D:. 55i

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