TI CoprS
NATIONAL ADVISORY COMMITTEEFOR AERONAUTICS
Ol GENERALIZED INDICIAL FORCES ON DEFORMINGRECTANGULAR WINGS IN SUPERSONIC FLIGHT °
By HARVARD LOMAX, FRANKLYN R. FULLER. and LOMA SLUDER
DTICS ELECTE
pwfi~c- toleoom mw swo,
&Bisibutukm. Is valij~fsd.
94-10501
94 4 6 035
REPORT 1230
GENERALIZED INDICIAL FORCES ON DEFORMINGRECTANGULAR WINGS IN SUPERSONIC FLIGHT
0
By HARVARD LOMAX, FRANKLYN B. FULLER, and LOMA SLUDER
Ames Aeronautical Laboratory
Moffett Field, Calif. 0
Accesion For
NTIS CRA&IDTIC TAB
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BY "
Availability CodesAvail atdj or
Dist special
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BestAvailable
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National Advisory Committee for AeronauticsI1talpita~lrters. 1512 11 trota N11.. Ieiiti 2;;, 1). V.
('Valt'" I edb ac I of C on~gress It ppro et~ I N\ I archi 3, 119! 5 forIl vt lit i pt isioli alt idI iirt-cto ( i f Ilit - sien I ifit st udyvof t he problemsni of fl igh t (U'. S. CodeIc tij l- 50, sec. 15 I ) . Its oii t etI irsli jp wits incitriztstt from t 1 2 to15) 1;. a vi
approvetl Martl, 2, 1929, andt to I7 liv acit aippro veid Muly 25, 1948. The membietnr ~s arei ujpptmeiidi bY thle I rt.i-sidt itStill stive its stilic "~it bolt collptelsalt iOl.
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REPORT 1230
GENERALIZED INDICIAL FORCES ON DEFORMINGRECTANGULAR WINGS IN SUPERSONIC FLIGHT'
BY' 11%it %.' 'acit .L, Hm %x . \ i V It F( 'i~t i t. I 44 I 1,1M A "I, I4 Hl~
SU'MMARY ~ 410 jit ioiis. Thil lll' 44 icllaili filIII iti [i* Ilit,' 51141hillo 141 Ill'
A1u m~~ it th dis prIt ,quei 4iiit or lt rtit i t,i ait, I f~lt slmeh' pendu'i t zed1 ft" l''l141t14 ll I hos t lllt e' v 41Ilit' 141 b%414'1114.4 ' 14.1. lan Mil' l'.,
.i l/ 14il/'/iI .514 In (I'.5 444404I4 4141'i54 i.. c form r~ .b. itsh 414 cI ..I tI tlpinil' I s I'l lero i lt -I i 4.a i t o is11 '.'. Gi linr IIint 'fl owtiii 0 44'1 /44 i 'd s'l p.v llli s pn t1 1 ' i.a iru 11l a als sami.1415 i) Iiini' glIi'l.i l'l a'jlil a hs 414 fll. lt'Itli
.1p'loli ilrti.54 lI4141 ' lilt'1114 o l tho coivr U1,l~i4nii- ''T4 h 't/ I itfi ei il whichtl, it ign .lb a' (ifr dimnion% t itsl tirit iila' of lItii-xot oo fr hi'#) vi1oity44'I4 pdf i 'd~iuis irt- uto /'jits41( a iIi~~i fwll 51(4115ou ol itis 4414 44r' i''l l. iv li fil-I. etiliti' 144 alrit ui IitIltlllrl t illi,
,of 1/441 41/of l iih j,4il'iuolc~l '.4r'. . Nume4,rieti/ r,-sul.s low/ Ab1e4 h itilllnib'r (if 41111l4'1sl5at1 i's Oiill. "'iiilitlf4414' fair- I hat'vo I'.1 c'ii
opmb r qotill to 4)4 1 .1 ( 1ii I. ol, 1.2 flo i ref tilt i' /(111/144 ils i)II tht 0,101iti s I 'll. (Illt ) l I0ila' 14tei'lllili'I biY a ''at liv inoalo 4 14-mal 1.111a"ii
fi~sl ti ii t hy./nt'l illt4 1( ehoird/1ivix 41,41 j/1imnrim, o/jrtcltion,,5 1"IFat is. tli&' 41014160411 fill' till, hIet hiigilir tialil-dillat'ilsioli4e
44 /44l'14ll/. 44441 111(11l~4'i4!/(14 41//Il rtof)l owiia dtit i'ls f oi.i ~It i s laippareid': 11, i4,.4l. 11id sivi li
INTRODUC'TION ta'4'illliqjih is ill itself Iii' no4 ilvll54ls 4li1Jl41'. h~il5. lIliiI14114111'4l
1114tt4'l' bi4llldtrili4' is thel c'14iliitt lilt of the laa'rod~tuntlllii force, s 54111itiil to tl,~itt'iltioll
till Wiing., Ilul~iegoilig sminti liut tarhitral'v 5l~t4ll"-iS' W411iltCorad-
\%.I,(' d1istor1tions1. Wheni Iill' will" Itspe'4t nI' is14 large (aw4t iltiltv .v V ', b) 1
SOwse forc~es 1114' IlsilalV , 14st araliVSill'a - )*v s m strip Iia'or. ill bu 1411 ilI4. 1v 1t ire t4 oti er p ir1ti.41 i f( TV1114''il Oi l411114it il 111lid
wi~v d1istortion1. Thiis repo)4rt is vo'iltl('l'rlla'i with liw-apll~)''t- 01tt ill at phlll4 E 4'4I~ll Ist atIII. ( ,llrliflt'l iia'l'.arli a 5it so f
5p4'i til weing lag wingsI( foi-(41l5 s till effec5'1 t , s llil' lifhI(' lliti'5 (-qia 41(11141 15'4is voia ll" e ullatilol 14111) nla lat ii' v lisII'I p1ti4~li
land. tit's full'. 1044114' Sil' v~' ofll ag Ilal' s he used.5 whic III lre4 timlt' 51411 11411i 54411(litlt ( lit' ) l l hi A ~ilS1011
Till, ~a~ctlinvu ized olutin fortiltforev oil hill 'vol)t' Il. folild w ll 11111pld it)Illeiitidia. N onditons lill- lll'I.
Itp c , 1).) i- > I l'44115n to" up14'l'cwh 41m (st de th flli l'll)11 w n s l 14141v-tl isoil
11~~~~~~~~~~~~ ~ ~ ~ ~ ~~ till'ra I.si1111' 111'/1114ons Hite l'l'ellstoftuch of ill soutin le'.vliv il ' 141441t i bliltilli-lill
(.I'It ( /istl, pIs tiae Slcdof th s lli. re or IS tis ll's lae.t il te1( .c/hi 4lq e '..jl' it i tll 'if ull it trilll'hl 5 Illil''' fl 4'rl his .'1 iie t- isa- ladw ill s fliill
11121t rs.'ll" N All'f.'.' th l' lbor tinvolved '.'kil 14.es F'llcl4~,ill.,irion 211.',141 54 0iad tttIv ln it e u tl" o r il t~ 'lv.ls h i
to pr v ll 111li rial abl , o ile f rc s in ilu 1 '\ :r lss d r ct 1)11Yi i ,i ,I Ii 11 c . Vv %.iI i ) )1 i d II-0'
ofsrfledeoilatols i as. errl eluh t r sllt mehdofdscn irclv oeqiton(li an hel.ro.e
RtE'PAIl1' 1 2:0 Vi . \.VtI)NAl. .t1)\ l' 1itl' ( I 'tML I I-'f if • lk F tI- \NAI '[14s
iill , .t l, t- a ilirpositi' ilnt 'grulIi tot ht itlim it I v pollt' i t Ii t il \ I ii l-I lt Iitl.nl I. \ ivs of lit tslt clid.\ m l i oll ý~l, ill I m 1-1liv litll' I- l(H~i• . 1.4 ,p p• I I,#, It tI Y Y;
I ih,, harni,,ic. ,,,.illa ln Alf I,.il' , .- ,, \6 11g".F i, ul \ tilt, p rit o ipad if l 'ivi- relik ill o f t i l i result, %\rill b YiY
r . 2ta ' wit " ki 'ii!t i 1 tt 'tgv
111114 c ill' t i hitIt. o f g till-ru li <'d fort e s. , stie it , th e e( .ilt two i' (iii J. jl ( 4 y
d i r ,.,. i l l ih r l luvi . "l i f t s tul i t ." . la n d i t \ i l l I .i r v i r t h all ` t , t h1 1 ,-1. 1 1I•'\ illu, aIlrell
thill tte lllllll of lailh or ritit lird Ito ,ah'uhlti ittA fhltI, fS, A i tt form(''ilutigi'd bv Iliint lro t r.'.bl'•t tli r ' ioll" 4iu-t.i\ ihi ftiiu t I liit .'% (Of.i d rlh','t,,d allcoll'- l,- phill formn
',(' AI'll I th l teloms l iftl'- ti hije u ill lefthtttl v t .oi
LIST OF IMPORTANT SYMBOLS
.| tI• ' ailio r
• l,, S~~ ~~lt<<' iftllit i l I,,•.i
''• , 1 U 1 1 Al~" itf- ti ' of witi g), It altlo
/it ,, q) ibtliit ful itio'l h ut ugt( t eq u. 1111,7 lt11 I t 1 willi g olftpiil i v re11t1'e ti
l ,-eitull i ti l fi llica 't iil (o t le e u (1 1 t15b gt').) t'ot t 1111i 1 wZlte tdil of illet
X 1 i till oli t) f til lt i ulli fill.tt t it s i. olif si th g e.d ' ,t IidgeAt td i t'q . I.A\14) . ) - • ,
li t c e l c v lI. lift W " ve~r ivai I v +. .ii i Iv
q1.,.S' Y.?./.; ('nrl'isilili ctidi itllitae, t ixu d fift ili vie t titis fluidiindhicia lift el<ltici i diue' ito hilth-tf-'it i k lit intillnlh
uO (L .4ie04' 1 ouitl'l ilAl f \ ith If i ttiill tiil i.tlltder of winliv• ttsuiliu
tindlilt lift .ovith'ient (11l Clte t 1 i lt ig fol r Iti X% 'ill .,i 1t''r i fla Ivs with t rlt igi o, clutti i'i " g f W7. hadi
itDitlS l[iSI•ottli iprsoat liie pi'll sidel lit l i ttg. zl'< 0•''lg 4 .
l iv lf t ii l l t f l ltl l w i t• i; al e d if e d lg e , 0t t i t (al . g 4. t: t , 1o 4 . )a t
'it.hilil-ilo lllt' vlilt cofvtiltitl, Tosit ive'E, whNn il'll rTiI-Oing, vd<ge' tendsl.' to sink rehitive toi ]<vidinv edlv, ,, llIl~llo ltli t l
<1,,5,'c .\N107) Al, (J.,, \,,- - "
1'ii illgill i l'tilth I tItitIoIig-1 iIn elII i lof'elit i ole tto 1 t
fre-i tlh-of-iii l i.k itht i ge r wi' hol't pie.ching) i l iloit oi f alill i ttk ilie ft, altl.I en fllight latih td
()( gI eiiitu lile o f h ill- ), w ir tditdlll l
41. g~~elaseraiei falowrIce eoreptttittg tdge C i itt- titaut friitttitinti
raidivinl i .liortg-nlitltiit q ell.iel..r d Ile to w in g, iof p ~ilh r V lit horiz lilill, posi-
iihhig ll"Illells red faOllout tile helidhil edIge foi. iivv, whell irniiling, v tg, li• b0low heiiolingit \viii roliittili< ilbolii its hildid li ,gl ,. (t , ,, edige' ralidillils
Of ',,, • i~ollrll lle, n sui<, lll'ii lifilh dimnillsion
oq .p., fl'evl-silretklll dlellitYlv
i- \ilg chonrdl vtehlocit lpote~ntili
.V) i gt li i'i i ndhdiv.iall forevi oi~ ,lhiciefnt (.'pe( CAIl. Ir !, portlionl ofl poht.J viltle ildl indluced bv so111-1.1z
f~tli i ill Icl asllti. plhilt foirmi
(h,</ g*lleitrllzedl indh'hld force~i coeffticient .'1( eql. sr "'por't oli (il velociiyx p enif~llill Jnlihe''d bly prl'seSice'
(317 ). I ofl sidhe edige/,(J.r.j~i distlJ. lilve ofl wing, ('2llnber lint, front .- 0( p)lalle IpOienlitill flille ll i lln lt ive.-dillillsl.':iolilil splice',
. 1 1. M li c h i mii l i b e r~ l S, i l b s cr~l i p ti .•11 h~lollillng(.illfficlitil (preslsure1"l oill Ihe' hiw l*r •.1ll'facv. reB¢' l(gionsl ill filI r#, ý phinet (Svei fig. 7. )
q " ~I m m p r e ssiule l i ' '~ i ' o i l t h e u p p ei r S ~ '1 -f a ( e., d i v i dh ed 1 ti p p erl~ i sid e. o f w i n . -- i 0 4
biv fre'e-strlenini 41vlinlniv, lim-ssllr ) s Jiglll l'Jly {it .(e ., s.o11"rv) plosition
( , ) billoillifliul vo'lictiv'iltll. tl If . ... IW e in i ig s o n ilfg r
q liitnson~s iilt,(i lithn .ei STATEMENT OF THE PROBLEM
q .iitsols it fptlil THE C;OVERNINGI IEAWATION'
I Assumling• it Vwill"',s Veltivill] mlotionl is of such,] lit iIIIIIIIrq. ~fl'ee-m rll'ini dvll ii( (IN11111 rsi' l re.ll-, P4 p •i llil Illie vvloc.h hes induceid in tllth hluid fare. slillill rehlaiv J\°,
q , g e l i( tl'ii z e d c 'loo rd in a tl e t h e ' nllig lii llde o f t h e ' W in ig ', s ivll id Y fo rwl'\ l l'd m lo tio nl, th e
Qfre o rs on ig to tt-ge - n rnmalize~d formli oif ivqiinioln (Ill)
ertlizel ('o01"diliinte q, A, " €,, " p:: - -ý Ct I | )
(.A RI-.I•LIIZKL) INDICIIAI, VOM K., (I'N IM.FORI• ING I\V Is 3
where (It ,,'. caln be itself as the ",tax rihlll pimrthld ditfirer l t ll If Ihcit dri\ lt i% v, ofl h \% ithI espcl,'t 1,) car'h ofl IhIi-, oo,,rlh~A -111ation I I thif e I hle l i ch " t L 'h Iis 1 ' u I Iiol| I tpplh's 141 i t, I. atc -mall. 1 1 \ 1in ll. iddh, 1,m 11 ,.1a 1 1 s, Ih I -.dhA 't .A at im I i,.i et erliiI tIit at Ia f t(. \ehvity pohiIll tll ik,hlell Ihe boat.' or will" ,'xlWrv-sho farl iv h oI ld llar ,.o ,ldit IW,, Ic i,, 1,-, it)ill q uestio nlf Ill.' t hr ough th ilt hi l. th .itxtl Ii itit il" lit , O"t 0t i l
lixtd ýithl u'.1 pect to fill- s.l~l Il l ilh tc dislti fl '+lsl xh' titttti/
I t foln jilt- Orrt. __ J: Y
ol'igiln. Fi'l-r com nclieanve %%\c Iplo i,' the. ,.vilg headil, e.dgeI
oil tilt,, ;l x:.- A t 0 Itand thl . tidl td'e till t , .l t il t' .1 Iltl.• I ivt'jtt'ltt i) ll hti lit t1l ll" d i, l lii ini tI-
%% ling+ lilt,, ILI it a .,talilt fll.%'% 'd ýIn I lhc I ,• i\c I. d '.I .. ti, , dic lll. t it iit, jI g i ltk h -lii t i lt i l chi ll i -It't di r,.,I loli-t
lpc ld so at s' lastltle ll I t- h.2dho vge lit's Ill ttt 'lltj For lilt, tiIltc. onlIilit i' too of i lit- lln. c lp/'tl- Ittill( . .xAlf, \%\her11 .11 i-; flat, M .14'h litillilber, kinld fill , ýic I)\ tlt, % i g plllu for'm , thw \rth'lcl vlhoil +\hicth .cr ~ +.
J +: + •"I'I' t tiiiiilm i o , alijt, - i i itt 'll i~t ill fivo' t . .1v\ il'lllti'liit : 1 )111i z s 11' c ioll, 11 tiiiift[itill t i' ai t t1ilto
to+ ha %;; v II I f r
10 - V1 \, i, 'h M h t h. ',. 1 at vlkIl l;IIII /lan lll il l 1, zit,i >O inte•, >' 1),
T ill .lhv .xprcvý,jol J Ilt+' is lledq .,o Ithak fill./ (Illht- m lvll• ,,
tto Oll- \\ill cailliwl.' lillv' ;it 1114. Ica2dhl• cdh-'t i'• tilw-rvll to• Ow,
ll;uhll-plllh lkiluh, o lit - vlh , adili-,edu,,. (oll-ide+r. I'l[- r\21n11i,.
llt' (il ist, /1 tot Ilh iti'\im .1-h '
Srop sv lscu . :Ili illfil t,. ,.hl~s of" siu'l'aic +halpv'- Ia1\; " Il,,. fori
%%\hfl,. i+, ! , ;Ii'- a Irll ili-1r.\ fl'inctiol ;Ind h 1... 11\ ,h flulit Itll•~ ~ ~ ~ ~ b.If,)III i till - "J'll~'+lft, \Ing'.s ca hil.,r o lir '~ l th- " ) p +.
wic .\ ithin i+ h l +'l, l: . of liII,,nriz.ed I11,,,+ . 1114. -ol ilolfor Itic flh w 4ht1 l to Ihiv \\hliu (,,1.1w lds ,o lds Ulpo lilt- \11h1. ofI
,c.+++!. . tilt- londil,• Ill :11 il t-h* \6i1,_- rep re+'"cillltod It l h , O w
I",-.1 . n.lit+ i ,. + ~ ,lrtt * • - +ll :lh*)\" V l'(1ll:ll1itll iý lh, ilt, +'
lIxt it" il•.I,<o l thw 1\%o+ -;pvc+'hl]
THE' HI NDI"iI CO¥ 4N I)ITIO)N."S i ,'i,+' ,r
T hv f1luid ch 't nort' all lo+ the4 sII'fjI(- of at .+ojil 1114)-ilug ii /+ f !1 I l iill 2 fri,'thm h.v.s thlid rfoll.- Ill, zvr'o. If the, <,qlontiol oif Ille For)I clise IIsolid's "Ilvfn,'e is r. ',.c•h by+]lr!~ > a" [ 2 /..11'
G i"< .;'9 -J,f 1 1)) I.|f _1`
ato l tln' (ý n is at 111l11 l pj itchin 111 11 liltlif•l'Ill l-Itlt+, IIlnout~th i s b lo, ld ihlr c o ln d i tio n ra in il , ex lp r es s ed 1 11 1 1h e n l l t ic a ll , v il • ut l is l e,<dll " " \ h i<.h i s fl'+ ll, w i l - lh ,. fl i Ih l p a t h1 1+itl +l f lllý of lh1v c<:ol'dillite sl\,V"IvllII d I i n( vIt (pllltl ioll (l' It-) s
, ,,,.I Il2f ;
of') +oxo ,O: C) y+. +, o •• " aIs .sho\wi ` inl fi~xurv 2. Ilhiiv., ltl (itIIue the, In gtit+l Ito Ilklt
S' , mlsid cr it th in s u r-f a cev 114 21 r Ilil t 0 p h il e . T h'l e cq s tal io n o f fli ,.-'l t pImth o f th e h ulld ill" ,.ll~r is
the, ,.llih,,r linle of this sol'fi'el call tOwn be vxplcssed ill thil/l-+il' .,/
( / l !/ : ) + : - i x yl+ . '../ . 0 T h e "l ' l< s o P e o f' O wl t l e a ld i mg + ' o f fh .l a t , ) ) li t- l t . , m til - ' I n i m ,. i st
and. a.ssumim, that thicknesss and liftim,• (ffccts caln 1)( [(01 1•?separait e I' l v(IIi IIIrly , Ithe Ibmo u 'idi y co Ii t ion for Ithe (.21 l.,b r a¢),].+ Ef i n e f le v o< l n es W lld t h e t w o s l o p e , a nf ' . s e ei l l l t o I l i - mq i \l 'v hd e l t .
+t +.; 05 , O]
(k 6Xm~;- OX. by. by o .,.t luvl' ..... .qHp" Ial, - m 1"tu
It EPORT 1 23 -NATM4 NA I, ADN I sO I 0Y\ (xI Ni I 1- EE H Iit A P.I HO- Al I1 I'
II Vt'i U 1It I-t Ii I I I onl' fog I- . I > 1,. 4 1i Ilit' I ipp444.114 V 'II ('lV illi~l It"-
I I f I Ia. l 't I4uI4 .4 It 4 )111 i'14t' I " 414 I u IIIIIII. 111, il4' a 1 .11' Of I t 1114' I 1(1,4
li I i' 2 lia to11 II4~l4 ft o~f'r4 I.ar al l I.a l 411 ,(1 11I I to' it'ai II Ig f4 ). I. oi'g I4 l .I I I it iI I I pI l r 4 .1 4 . , I . it ,itI I11
I111 ,i 4'4f1 i44 I IIII- 4It4 f I w-, ,~t l4'14l11 '4lir-4'. o ,l i t 4 i wi-41
f4I114 F I. 14(il.I t 14 j444 lil ts (A1 I''14411 fIi 11 1 I 1114- 1l
to f c 4 14Iig111 4' ill, 'l I jI I II I~i i'41 ' aI441 I I I 4'I 4l I 4 '4 g 4. I )
" "14 ,111 4 I 1( I tI%%i I.f I4 I4411 I4 Ig I% Il1.~ 11 41111 114 341( 'l~~ 144
l4'Iit- i 114 o 41111 Of I144414111111d't. 44. ,I'4'14 I I w 111141i i I'll nýito I Is~Fill. ______ w i iiat__11 1 fill_ t II I I- I I_ i I_ _ _ it 1(.q I ft w 1.1.11 4 )1_1, 11 lilt I vo Ill appeal.'4''1IIIII I I
it It 1 -6~~~*4 11 , tI - 111 itgi((ilI I~'~ litlI n Ii. I. Iu 114 I lX ( '.I p1Iil1I440vv of .111 gbl
foll b1%4 lg('ll ll4 g444i'I pl'I 4V11 til, fl~l.11g'4'' I ( 1 41 11) Igl''14 4g 1 ami
III[(~~~II I Iv lv i t p a ' ih o~tik'(i kld o 11111 h O gj fi l '- I Im -i l f fi ll, o 4'44' %%111 J l hi4( ll d w1 q1g ljI4'Ii c2I ll i'4 l k(l v144 1114'I'lilvi lit. it h . t114 44 Inai fllrledtla i v' '1441I4'. Its '44141'Hl gl ll-,, ill I 1li ' "44'.'io viiIlo'. The. fll %% Ovg erI 14 11 14 ill -1g1f I'4g1411
t'o iihllt( I \%f ioI'lg 115p -sil 4I4'$ig'4'41 Iis 1114 it11'114t1lt1j111'l(' I 11 '141
slot~gll'x Iv414j 1,4111 144 v44 Il 111i'( ponloici'14 Ic p i e tll 11. 1
44'44'l'~~i list Iwill filll flilgt, lgl-l. Ofll4 1114 14lit~ Inn lilt4 I- \\ Ing
SI'g I I I I: I I. if .1'4 4 14 %% II" i IIIs -I IIi i I14 Iii~ '.., in t '4 11 1 %4111 o 44I illI
I to ' i I IO THE POTETlA RElliE Of IM KIR(HSflt'I'VP'H. FIIRIi. Ho0 v. it
I~~~~jglli'~~~~~~~~~~~~'l 4 14''5 114 1g 1111 f4m(I 11'51'l(( vf '.41j(l 'l'h '44 11 1.,~l vl'(1 (j I Il I 1i. s1lI(4. ex epti4' l ' (''if4I' 1.- 4'l
111' j1414'ail lll41441 44 til''.4144 114'1141i(gl('l. r14 '.IIV' 1I1(ll 1114114121 411 Illilim i~ '..i-lrer '.).he 1 5I'.e (lll lll' 1111 d 1114'k114'5
11111h'I'l foi'(4111.' 1444 4'g's j 5114'(l4 11(' light51)I''( i411o1l'i'-ll andjllilt- lpolv'ljflj 1: 5111114's Wl II (ll '1l144 l'414.j4' '(11:'".s~~~~~~~~~iijt' ~ ~ ~ ~ ~ ~ ~ ~ ~ b soht lliil '~. ii 144t4e (Itl''''4(jj' l- 14'jInvijlods'44 pljslbi 54l1i4) ths sjiillpl'. vto ol4''4'4 141js314 "IlgT
if1 6, %ov -4 il p-e cil t 11ifvi a ploa h iflli0,
6EFNE]+I(AL}•D INDI('I.L HCI.(", O\N D.Ir()tOHININ 0VN.
.Ik Ii ',teald -,Slti* \%,ing• thetor . therl'v 1- 11 ol-IIl' IlIlh fi~ll.> .2tixn,.-dv J,, 11h " s I I.,•• l ilt I'Vihltle If. lic lvio '+l v pl. nt h Itlll toL if +++- • +. ,+.
dlist 'ihill it'll (if soi-d '+*l h l rit '( ll'+'' Wlld doltbuhh ,t ,o+ c'r +it _I , -. ..
Kirc'h}ol.l, wlll someII+ ofi its. Ift-ro+d %lam c l ses• 211-.-+ i discil.+,,,I'lill retfql'rellc'l' 6, K irc.hhoff's re kilt is il m ,lini t-th l.\- applicalen h."
ilk tl it' I il~ l ofnl 111sr l l'tll• t i ftl•• iv •+}ll,'-sur act p olel~ l , " l it-it I•, \• Ik'I - -
Ilk, ifiI lp ,` nlA-t) Ilk' ripr et'e" 1'f ),. \ o ll cc h ' , h , I, • I tll Is. , .. .
Is thw v'•,. lit n',.,_ioll, l. Ill..and Vl (of" lig ,• 4. 0
"I ' fo i'lc
\\ l'- -I- l'I /' / V -ll,+ 0+: . .\',l-ti- ' IdI.t11 l'm,, -f ,,r r, '•tf,ý , 1//I ,, o, 1P'H
•"''''"" 2y -- r'• ) '. lJ" l'dt,.+ W ill' t|| hese" correspon[)L d o11 114' subreg•.ionsI //" ,;. Ill,/ ' , Wll l lJ
".,,iid..tilie.d ill ligu 'r, 4 ['sitlv ' •.< 1tl•, th %%v 4-1,, lilt•rn ,, l .
Po.tent, ial ill, ., lI sI Illlii
T h, '. , ' ý l • tickv w ,. i i~ t , I tIva Iv IlirltI the' t',.tl ',h'd valutt is it ilt - silt/, _t I t.jI .i .l 't f ki';II k,,,
iitifor m Ii. t i, t -tl'l.t 'i' it)h. the i tlll •tlli l- i iht, '.lliit •• \ +h-,lilt
air ~~ ~ ~ ~ i ittisc alli it-iitltil retiit'le
••.• i•,•,- 1, II1. til, i. l), ti ',(, of t N , ,li Ia ,ol l t1c., g m it-Il i it four-dil ii, si, , .ll i -i i ,-,..h .rIhIlX f'r'om lit, of A lhi,+, r'e.rioz• Ito l other. ( ', .ide'. Im. object , of to find forIffo'. ,,xI1Iphv,. Tlh,, ot.'lil+l~ t f "r for+I rt'' ioll _III. dhno<ted (that' < illj sfc, fl,f I'" r .o I tIIbd? Ir%, vo l hii s i .l~l~ h Iw p dlillh, 21
,.!If larl oft fhi l •,l- bo nd .\+ of I11v avo, ,list . pliil• formll S, h t"t~ i.. h, ,,~ .. .m ih •, ,t , Ilh " 'i ý f ', I I t 1 1 k v v i n k ' i- ' I I i t k gl I ' " b i - 't N N e v i ll tI h e te l l l lI I li o l l o f " l h ' s lp e c i l i e 'd i h ) v q u nl i onlh > " 1 2 n • , a n d 1 2 ) ) , ( ) 1 , % i ,o , 4 , %+ , ,- f i l
ho.lidi ...z vd,',,,. .viT7.an tlitlth, vxpr'es-h,,, Ill% Ivs+.'. 'o l:lllls ' 1,1. tl spltl•'. o}f [non.'l dlimel'lnsin -il edl iZ}'ll ll' kill II• ,rlbitl'+rar ".n , % a Xv'vpI I lhn I it IlilI 'II". I "l i~f'*.v 121 [qla l )+I+ Iv', ill
J*" X+,-. .i y ! ! - t 7'j Atli .r!.Y - h Perplillit.. Tlhen. if" At soltllli tl ill thi.- hi.,01h.1"(Ilinvisional s1it"v \% Ihih s+it tiv.,', v,' tllli }lt on 2nl and11 21, ý
I (I h ri\, uiv ,- Ihliv ol11v." h•o,,mdarll . It I thlli." t. wf atll Ithe. IIis- ill tIlk•, .r.y.. -',t h 114. ,.11 I..b fomund. it ri,~l.-pn- .lls tf, w" lic, ?ihl Ii-t m blllh t -lll , 11l1lt . opi ral l;l ill All| call" " .( ll prlodill-vt lilt ell',, t t ionll i dliffl'ls:io~ll ý 4-4IIIII 11 +'.o 11 ollv c ll'tlll 111 lhl- S41111 loll 14) 1)III"
lit fill, Imillt T.+// lhi.s bolm harY i- tilt- ,llips, probhvil. T hi, (,m'1111 ,1le.Hz,.+ h, 11v ,elhod ilf d,.4- .cil ItI .,ltot obvio~ll.'. of v'Olul's , thalt s.lich it method 11,1141, to lilt.\
Y O n Sink plific' t i<t I: I)II . \ i h 11 t Ill(1) (.1. 4-lon ' ,i•'• ,f - I h+ if o ,, \ AT III A t!(.tl~~g .,',,,) -I,-<!! /Ilat ion" (4a . m i t, f, I he, l,,\% spIre.,. +,l<.h if I.,s'ilili t.% h ;.h ,\- ,,exist.•
wher.e Tlhe.re,•l• t~arv %% h. x~• r•, v io~ll. itl lpliv Iti+}ll, o}f Ihis..lls - I . : ill method hinv'e pro've'd to Ibv tiII(,fnl, tl -d " ]is('• -•t of 1114,
";'+' ,d I'"= +dInvilhlod. Inlll.thom -1 hi the hn u'odt•licliol, is,.I.-b 4 .Issc . A .simple.lppli,,,Hhn of hliz Invll.o is tlht- de.rk;lti,.l of Ilhe vh.,il v
If Ihe. point x..'.i lies+ ,,vhiin the. c'.ylillh'i<.ti wave, from• ilhe potvillitil fill-I 11 soII'''ill-c +i at \%,, dilill-,is•hlal "siII+,I'•,'o +' l h w H•
Io dlw, oll vv,• ,' .. !h t i+. te ..x- t. the ellipse+• ofi v.plalliol +4l A A) Vd. Tlhis. l:,I, tI inld l,.ld (which am.,milts 1,) :1 -li,,l fislhti l,n•,.,rili'ise.s only partl' of• the< avol,,.t+l. pl an. d fo+rm , lhbe rl'lmll lllfdl. l l' Sthe, .tl) oevurr1'lil• lit O th.M a'h %%11•\4.) is+ ,.a.v h4) deri\v if' ,II(.h,.juL,Z bol.+ idhd bv so 11114-h o+f tilt- .ir,.h, V,<:,s+hhvi,, i t '..d m r+ n fie.ld with It i lilt,. of so..•.v.'
rlorm al to ilh l'ue , -f 1'v si lv l and ui form•II1 ill st 1-.1141 . "l'h,I r J.t,- - y,+ -- / f I;41, t w o-d ilirm • i,'t mn I lih 1 ll1 14.11lll (Ii( + tl llm k\'(' fl )hI ,,. im , , iA l ,11 .4 ll I(A
II -it.s (I i i m tI h e \ v i t Iw fI I it i e z e r Io . F i m i rt l 'e 5 s h o vw s I h , t i l t- , ( ,, b y d v ,• .+ 'v l l t .I In otlher• .xallph lvs .Ow ad itional diniv,•s+ouu i-s lnvilsll-',,l
lim its. f r ll•x't . three, tylti,• lr,. lentle•d space, i•s O 11h11 11,14n,,m+ll lln u lh . l'. -liotl di ,.
R ie,z'., In 'th, l •disulss',, d, il 1-14 71 for" o., vingl '(_. 1111tioll , Ie)
(ii • .1 >x,> -t illustrate's th,.,, 'Ls(1t
6i~p IMPO T 12:--"-N llu L D\ IsillY CO M NlIRTL tilt(h AP.lRt NAtIlCq
1110110(l~~IHI 1`01'~i S~kLlltt voi -ni. t-1tiltEl IIv s Il. lca.
"oe O14d , il t I-qu.r All0:dir lit ,,id rI hi)o , fil lc o~gill p -iv
fill -- 0/.'4t(il II (1Il m Iii I0 t 1 tI1 1 1 . ,III%
lock. jHlt ittqg 'it o m hl, ýIIIt ft-/co fr
at11 1s t 1121 '1IlI ttoI s o I- llttioIt 1;) ll I tis spi t.ar M c, lv luplil orn T e vo ltIliilI ol f ft
tINII IAII~I INII if. II 't III. i %III 1 . \I t\-it INpic Ea. 1 of' v. I 7i rl611 ,tl - 1 allo ol
It)'~ ~ I, I l~l, h- mit hil, I % .li I I , dI I II .*I I( 41 I I IgI lu1. 4 lf 4l~ - 14 1 1 tillI i I I
litIX o fa p a i l.i" a l ili.lf lu, o lI
il i 1 ., % 'lp~ ., ni In lll. 'Ill 11.11 .- du . * o
N
Y_ '7 7,
\ ~ . .. (~ I. I0 o o lo l- - 1 ilc f i \hc - I -
It. %_aIiI______of____llmk_______\II_______hw_
he rolldi
Il o ' ilv clt II1 11 v l i . ilf~ l lili l N11.Il- 1*1. Ie v- ic 4il I im o ed 1111 1Ic o dil iN 4
X'. ~ ~ ~ b l Il Ill I-%X I t1 ofl t4 t Il 111)1IIt 11 It ilt)I I s and 11 i t , 1)1 It 111 it I
co lI-cIl \\ tI it .-.il rý\ il il a ll v s l
dep1l11g1 ol t~lt 11 X itil- 1.1 till t Il tlit e IX 11114 It ) i ll t It cot~ll tal~l
phi tc. Fh~urv s s ows lilt Ik di~ernt oti bir, \ vall. i-, Il, i, z
I.l' I lli)t 1 l2le Si IiteI\ i.l AIAle 114eIN M \ '~ Il I I 1 1 .w1% Ice I\ if 1 1e
kccc Ilk aiv fCi i~-ile ,~ lifelg 1li' i e.-' 11 4i'ii I'. IF% A\ ee el~ eil -I- 111 .. 4111111 fill, I ; tlte it peeciiI -
Ie-eeL- eeItl lilti t -t 'ellia l elm l IWI)ielle I fit f-eu ll- I lel iei'C C ee e:ei e ell~ec i'~I% J ileIt, al ~ eeei~\ A ll e tli
1 C ,t Ieie ,ch, 4 l .litl lilt
ee IN EMRF4 1,110%e leclelee cHElCi U c\-l Cos.el
ci.- ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ Te Il, -ifill-eglliei 1lri~ ill l',l~,' I h.e" I eeL l11e Ill %li.tee .I\ ..,Ie tel I,- keei ýI- ,i
Illii'- te4I, 'ci ll l 1401 17e eli-lel] liii.- iII4 uti le- 11 A. .lIe il k le-l. IcC eli el- fii t wleIieee. -f t- ki t-%e I, -Ii l i l" Ichit
le .- e. -I. ' 1l 1 It 1`4t-. to.re Ii. I l- 'i eaelt~ -'l (Of 'I t Ie ,I. II f I lic -11111.ti 1...l 11-.. \i ..- 11 .11 11, I I 14. -fe Ie ,- t11 II ti II I q ic.i I wt l -c "I d I i fe, I w I 'L' I I eIf Ifo ii. lr 111 -h n i tc e elitIl N. :I .11 t 1 , \ " F t e ' m , - I1.i I %1 . 1 o
frill - ltc Ow \1 atl Ic - f rol ierI te- tc' r1 Ow fie -ell,, a I .- Owe ~ j~>c .1 i I ,1 1*1 lI '.tl :1 I.,it -hi I\% I~ee I ree .lce'e II, f Ill le'ete oir IlIree. I
'C J lect i li Ife i2-1.1 cI '.1 e'l-knieeI e i e'illtf it ie-r'I Iii le-ee
lec -im.elec lift ii -"ltr i' I liei % etaed lit'-t dci'.e'le4lee'l b. F ml eerie .'~ee-eIII \ X~ it i.. felieridI hoiof Ill. letelticeil el ejeec 1" ici%l~i 1. f l Th. 41 li-tell if iOw t;'. kircl.. (, Pltelhl.-treý *..- It ~ ~ ele lifieti.- -rel-fee.-' *:e e, l~e an-C Ideli. '. " I.-cd ;I-
IIII I.d Ill li,_ItIee ' - tte f i. , O w eecilm Ie'le' rejee'.'etee Ii.'I 1)ce' 1,ei e ~llrl
e'eI I II lcietee11 i. I F 1 4 if , I I " aeie I 4 lea c-il a e'e Ie Ihe e1e cc -, I' 1,~ 11 Il 1114 111 .- I I a
ci i- lltit I . )i ee-1"iil1 ' fill Iee1 'e'.4-I--n rl%''. l ueet, IeeIeIIe to. - C %% ec I ~ IL Ii ý lice.' 1 11. lict el-t 'rrel ie I lie Ie1ee 111 teo eer I Ihe q.le -t IIIIIII-I)II% Iit-e I 11 1j/ee . I / q
tole I. . I- ii i I 1 1-41 ieete 11te1 7 1 elr -i Ie ere leeiete I cII1 1 [ict I teo h I II I',
we~id 12 Hldne
I1le f cee. f I %. I I t 11eC 111 Ae I-jittet Ii Al ellctci
I c .it, %~ C" e~e t- It ltivilli~l eV I I I It Ii eeI C
l.4-t I- lur.t Il~lee'e't e'e1 ilttIrer~ .It l m I 1 -11i 1e Fiet )--".Ilect Fileleir It k " I III Ii lee heirril 1ill r -eel II it IeI I feeill ce -I el -1rite-I I l-
rrteaII"t I Icitr lifiteenu -tir-ftwe'e I f it re'e'Certe tirerr- C .e lilt %e lieu c11t-0. .ltIIi 1-'%' ICC' it el .f I e- lee1Il 1- it it iefeeX' t'.eetiW
e.le .- c.re ee. .e ee .... ......
4~ .lel t.e .. eilei fet..e.celelIe ei r -efe CC II e.e. . .. . .. .-
_________________________________________._ .i tie -eri ef .e ...rt..
.~.. .' C.l .-
X, y,
lite 11
o c la I IF If I e'cl -t ed" It1 I lie' I I it e . fetlilt MeteI.W ''Ii lit cl ICC
E- c l lif o 1 7ee m eel' t iec' -I , e Ii Clii, I we it ie' NI [: rci- -l Ic at c-e 0ic A'i
of I I p " l .tS
11 - pfr fo- I''ee~'c' I' c Ii toieite %% In % i illIce. I. - I tah ill C r- lit Ii
[-: fi Iiee Ii it 't - I~i 1711eh Ieeteei tee im lie mici p -In Cepc td-'.ct-fce-c' ~ i tiec' 1 cetece e ,f It-cr tieel e vv It- '1.111- N% 1?, :e l Iti i0
I-c cll 'C ~ e'-e 'C cceli.ree ''c c -c- - ~ I~c,, el tilt ' I ll-- i ; I it li IFic NOc-l fee11 tee Ill, 1111f). I ' eec
Iz [ ill t it, ?1 ..:) Ill S . , 1 ;Il '. F I t'l
- - - - - - - - - - -- -
-- -- -- - --- 0
N M2 Y,
I"IKi II -i~fr lan f(im If I mi i rvi , i, i~ ,0
It u,1 17 111 .I I 4 ...- linHmilv w lilt-2 III,\ alr ." l l O w22 22 r.2' ''~ 2, 22
:12121 1ii 2 ti-j 1111 11221 111'22~ 11114 12 ~ ' 1122 it~ I r IIII tI II lI I I I I I I I it if I I 1( -121i' A 2 2Iv .I I II ' I I i. 22 21 22 21 11212212 atl 11 -I ,'2 2 .2 if211 Iii I it'. I Iii 'q22'- 1I Ill222 q2222
If I2 hA 'I. T lw I/2. I'i' IiiI' I I2 22' it2 1 1127 jI.2 y 11 n I i- 2III2
I I. I I, I2 222 . r .4.III1.- I .V I I I t Iit r I ' 1 2 2 11 w_, 111 22221 \1121212 I12 -I.'' r2 I..2 it2 -it. I.42
111'. 1:2 \I2-1 f.. 1 -. 121' 22 I, l-I 2211 121. 2121' 2'122222212222of 21w1
442222!' - . Ii t 11 1!,' 1 1 , - I I I:I I It I I -I I I*!J- i 2I 2I ( he2 l cm
h\ ~ 2'. 2 :I 22 d 12 "22 I22 II hill 22'- ofI o\ 2H22' I' i c \%' II ,- j
p2a it-l il tII 1 propor 4'.il 222! I-I-------------------4- Oll
!,l22/ . o - ill.\%ft- 2I1222ill22motion % 12t'tif 2 I I Ow itt 'iI- voi2
di22l t2.ha 12t Ili,-\ 112 11 ' lt''"'I I':I l pe i w 11 *.lplll little 12 ' I'.42222
*l i .22 '.2 212:1 222' InI 111222l.1 I 111'. 222 jiii, 21 l 2j2114-o lt 2' pi 1122'. 'I'll'22222 " I .I 21'' 22 ] r [i Ill, 1 .'22 I i lI - l )I I2" of 1-4 222'2212"2, II' 1 12 0122
I 2'h 1 c21l 22I I k tli ) Ill of. liti'"i12 , 2I il 2tl I h \ ll- fo \\ ~ I fit- If
II'll 1221 1 H il[I - , it4'I I . it I I lit, I 2 f "i( 12. 2 llit-' 2222 212 All . 222!it f2' 122 1 4'. Ii 11 *1
2,. 1'1.22! 22222iiz ti l 1 2221'in fi'. 21 'lio.- l voo221'12. ofI''2.2'
Ow 2 \% Il,! 22 112'' '2 'ii hoI2d i x1)\ l~ the 12 'hw fore oli .I
22/22!.l 2IwI 'vl222~'~' 2vIr~ -1j11'2I'1 222 a aIlILt'.422 T12'I)2s122 wr22'.
122 lf 2222svll the22212''2I d22 iffe22!vA221 2222 22ke'l Ilfc hg "222'f222'
for2 Ill] 121'"11 221nb -'d 'd l o lfll , Will 22t 211 21 I 12 222 i ft ll 222! '.4122 .A
2221 2 'jlit il-,O ":I2121122Itp A 12- 1121v 22 lop 12 of 114 122't'.'.2'2'22 \\ill. A221 N1221'.2 2212-2-222 iniari~ '4222 2i222 lilt ret Iion heIl\%i"2 'l S"12 t?[fi S222
W222.2-' I2xl 222 t12 . 221.k '.vf 1122 l N, i. r ;., fliv /122 iti plan21'.;2 J2(1 22/2 II 2 2>1 i!'I l 2''2"2'1122 .'22 r "'2 22 2
22il ff21 i2 fui;221 ou1l22 S,21,l2;2,2 1121 O'A;h- *2/f f 2 ph22' h1g. 12'i 22i- 122r. 1`111f IO IlPL , I
p/a ;1 ill A 1q f vv 1,1.12) it0
10 IM~POJRT 1 230-NA~rLNAL ADVISORY ('ltIM ITT E;f FOR~ AV1,10NAUTicis
lilt- I.,14ce pi aii luaint~ti ) form is i l ilt( lt Ilileitsllilt is filtile sirlopesi'd
il it , basi. lioulw 1)[11 orli O lti l'' ' lcmii,.q ti2t,
clis 1. IlO byil" 'giv l b.% dw Ii t- g tlegr l i-lI ill hii). i b 101
ltit11 1 -1)) jijlil I ii i t tol- ilate i ,Ig 'I II fixed o~il tilt wIi~~~ll," 0 t1.11 ~ tl li iI i~g441ii \ ii.iiI" I
I %%f fli0ndvita i ns i 5a) il 1 b all.( dt-/.:r a 1.2 " e tv I 22
e(Imitimis ~ ~ ~ ~ ~ ~ i I Ii iii Ill mill t 171 1 1,% hen ii. ip k t i-i n -/J7
a 111V lii lisin. rgoi s ~~jilNIi el for .ndqjjlj''' vpIlf1 llId.a t '1 9, 1~ O
I e fo i iail r ll! v o I h % lIng.r. ¶14-ice vilil1n (21
____________________ ______________ 1111 all- iuilm NNlkIli- lii it lit su e so i Mist ital I% iFII114 ltt
til 11
'411411 %lilll li i t v141. I 41111 rit l e tll il- \ w tn il.1
li r e o 41i.[l r ltr it l)e u n _____ _____ fibrlll ioitlt Illil l.- For ii Ihv pii l4litili -1111 4 14. IIIII II 4c11% IllL ,i t
aliSntli l, % ýilb ii ti-v v% Iilttitlioll of pVii tji ol oII~ lT ll i l,11
moi on IIflri, iIlh)r,-ivd Ili dllr-th t i II Tn, .%% ,~ il Igs n-h,'-. It., ll- lrlrein 41 1 fi ll it l-A * i l ; I %% l -ng .. t) f it .c I rs-1- 1-dr-ss :1 ref-ls-.~ sr-- Iisr llhsVIL s.tsr t uig sI-,nl- ,- r:rusrnlun rr lk 5, -v-r 51 illl-II rsi l-lh rl-lflt Irl I - s Iil-dI,) .--- Ia Il Iir-.rI 5
ri-I!~~'ý V~ r 2I,,1 wit) r1- 0 -10 L)r I,,q uI .- lll.I,
2 oý Y
If the fivei-inlodei frilqueutvies ariv j\Jpjiillijiltak. jtelcr- Fjill Ilui, eu . It it I, tOw geli-raliid o~i I c1t1` ore t-etw~it Lit'l I
iiii i Ii (Ivt. Ia t j i t ktI ils suchi a- equal tI ioi u i4) 231 ýg vuig tt wiig gu afo Iit,
Iue lastes, e 4 1t Iit tit 125 )il;)J1)Ivis5 to (uIt gt I -pýI IIr a I I,, su -true I ' c - /ýf, tooi 1ft I 114 s wit I I I % it I II II14 IIsI Iv . Vuiatl1V iill i ithe iap lication (if 0 ..
vti'itimi 1iii 25. mw iu ises itiltic Ind fii-4 11jievY 0,, (of Ilii fivet
Still pe. Lejt ils i)xa iliO (Ito, gn-jlictliize tititi terll) ill eqia-
timi! 12l',t taking. rowi .Unpiiiivit.il', oulY tiIvin iif the( stlo:'' if i. qu1a Iimu *A vaial lt, r itt 0
(,q, PX Ih u dxfty 21i) /I V,' I
qu,5 )' It,f , i [ J'ý:
2q i t I , ..
h 'o-d ll 14 111 ha. J*~v beim t it li... Allit (2 io''liAIIfre lljfji'j .('.a t'ii't h '~iiI
%%Ilile ~ ~ ~ ~ ~ ~ ~ i left q,.), if tit loa ie wiogviti correspon aloii to till THE ilNRAIiSE INuSA stjiiwi C
\indjw a gjifj'rtimi jl v prvil.,sctij'olfre(j~ lijlt i me b l i tdi.'ktjcidii' It f igiri' . fOii ll til -other ilillI. i sv tiiitviiig1 a Altti' iIlfi ll-mj
ft uI~i)vil: protl bem it wa iofi'ec ingi) ll wii i l" t I) ue vir. .:" I sypestjiic
I,.pet(17 cailh bi ed carrt ied oultg Iif 1)rde. to11 cooi v iert th f reOlt
h~t~~ qJI ( ' lb)[~ it IIIjI/ (2.S ) gtii dzc f((..( oriitae stfto he ilier"W. letus dfirstd raufe reqults
C C iC the0.. We wil I sit sho w t i '. D,: I lset va isti call bi iue 14 t atol S-firon, (he StJ l . t iolise to the $3./da -%~i 111111,es
T e 'ljiltit vi-ica of t les iiilY istritij'5 pro le re rseie ) wit h eliitioatej41
ijtilt-j'i tof )lpplv ori iiilv iiotiiii tniot~ ill'ejnit jiitiejiisit.lt.
Nqk% iti -4-neraized f i. '(ivi)ld') ftw e li ef. U il t fi ll' b4h ljge de file it hoe-itl a d l sid il ll.( ial
fItIto%% Sii: lil prhble tli %%tIte is01 CMIVýeitIl to ait w lt sioolli
we raj jei
isui fits lea tiu eth e W itg . Iilt otrd eri is as-v
-t t l , e u t
.IW q, i'1 (' ( (3 /0M) 'tills
s1,14 calui tate l ofith es ii illanu i t tivit, iiiilf t illeei l ab r tdessfli~ Y r n f r 10 4 ,1, !
d T425, S (::T [ -- r~r ~osf~iiiI~~l 0txr '~rtiaesiI. Iojie >
00
12 RIpecuT t23---N.vTttN Al AI1.0Ht) tttINI11 I. F:v e %F.ImiliA l T'll
Thll inliviall force. coefficient W" 1t' i.s ,htlled t,. I'.11 l,, s: ,lW1'4J... OF (•,, L,'o
. I -I,} j-, ,.I d " l I t • • A T ill, l ( 14,l t a-ifoll h, of l-I llll~l I,- x % lla. Ili l it ll diial or,.i.nt 1 th1.
ti. pi .,tt ti, f s.: l t the lirt part of" thii r',pttl' lilt , disut f l's i l i A". ppil. i i\ It. (i',-.it ortl'dm 14) 1ranl'fer Owv a~xv from tl t, -;e t . lS)4i\% tl ill tl111r1 I sideralbh,. labor i.. ll oin , 4d , l i i c .,h clihcul~lit iol . lnd lilt
I) flike fllol'q 'oml\ellient~ setl of ligulre 14..so lhafl mlode .shape•, a tlllpll t~ 1\z I•. lla • In lit.i)d ,,o v,.r l-v*illl'sioll forlnllhl•~n 10*\ lllelilll'
111,41 M.llet Irit" ol.Its flu+llelll r'l it' aholl I Ihe wing's slpat|\ist, of which c'o'rlaiti dt-ri %lit \ 4-, 1` a, Io lit, l'ec atli-ll nI. %%h r in,,lf. COUhld
4,.,,uhtr hllr, atid til-• fl'w,ve vo.lliv.ivlt denoted .f", v'.,l he if,-tv.r- bv, expressed it. colimh ottio , (of others. T'lhi-; alhinil~lp %%It,miied, \%a. proceed i.s follows. First, till, hoadin~g coe.llicien t 'es.4u and .6ide hd ell( follhwing rv-.uh.,:
l tt" it•'•sing ill Ile fl, s.! ' , 'qilvi with 4 )%it"i s.h gisht"iVn b\ Ittnsidtr ioelilt I .n t islt l'lolla I thlt J. il 'it-gral il this
e(qull tion 1).\ i l. till,
,,.1 (( r))' ,, -. ibiT.J. .,. It q (..
is Ob•la~i 'd,. This h01din~g ,'ldl~ilit~lt ,'al b V wri t te PII s at0 1pItI A
IP'
.Not's ith' (lt18 111 I', f" is t'tjiit'd il telt' X.4+14 , 4,,t svstt'm a1, F: .- I "--F' ' ".,
I J., -ill it If J~j or ze(cjil 1. Inspec)&t'itcn of equj ate toc (371 shim ls thlait ilt'Slit satme rlat ion
.) , 1X Y• r . o l
.- •3 2.se,., _ ,, __ 'tt j , " - x<'-. 1 q holds for til gem' ralize'd injit ial force ,'o lli'itntis.f,: tat is
t~.1 APl'tI ~, 'IX . -1 C, q+,,r ,, .
d2- ,t ( ,ly ( f• .0Ij(9
"l'h.• astintgra rn I wLitl'LIasFrom this relation|, it is -wenl that onldy the for.es• F"; livedl~~r-.++r . "• -+4- |1 ft l I- A•l,-lJ\" determine.d Iy in~tegration|; the forces fol' highe'r Vf-alit", Of
..Li-- " 1141 " j r, I" i .,. lex I ,,•, l lit found, biy ,combnation of result for elif-fereit values of tie't motd' shape index j.
.)' *-1- ' (--!)' " " - (- n As a simple illustration of tilt- re'su|lts prese'ted lso far, weS. ... . \J ('anll 'alutilate the indithial force derivative for the cases I-
V, ~I- a ,,=0. j= , I . TIhlii ease' j=0 corresponds to the i li<.ial., . ly , I lift 'oefficient for a flat, sinking, rectangular wing, awil tie
c 'ase for j= I 'ot'responds to tilt' indiiial liti'hing-nmolie•'.l
BY' using equation (36) we find coefficient for tilt, same' wing. Sinte ity-!l=0. equation (37)give's
2 -4') 1 t i -,' 0 . f,< k, ,
(\ )(0"-" 1Th1s. %kith jO atile([ ide'tlifyving -- V,, it, angle, of atitaek(--1) Fjl, (37) a, one', fines from Appendix B
whltret all fort'es are, response's to ia ulit itdic'ial disturbance. 4 it--) I,, + [= ; L I - : I _tt, 1
Note' Illit if vy'qtint O (:37) is applied inl tile' case of a wing (14m I"' i f A L ' -2 i - - M1-- I,ailite'-ere'ed oil a wall, both ii and I mmust he even ill ordertit sat isfys the Iiouih ary %'onditioeis of relhle'tio' ill Itle wall. LeS-s + s'es
BY .-upvrimlosi lg boundarv 'onditions tita[d their result-
I l ,, , sot1 tions , ,ti,.,le ' fit .,. h ow that tlte Value' of f'gi'vsell hY eiqualion G M17j is valid for all reduced aspect ratios ',,". _(I-.Ilt,, J -+ E , 1 1+ 21 . ,- O5.l gi'ate'r than I iin spite of the fae't that tlie value of Fl* 0iVe-' l')y '(platioft 136). as it stands. applies only to wilitgs.fil." wil,'hl 3.1 is greatr tlhan 2. . f-- t
(GiVV+l Pf,"W,OltW'1, 11e (10 ePrnilM tile, II(' genralized force ats-,et'iatetd with tile gt'-iralized c'oordiniate' q, bYl meaiis of the' 4(1 ( 1- . , 1Sle' rpos•it io ilt-e'gi'l as illusttatd 'v bteqjuation t34I. * t-- -I
A
4;V.N.EHALIZED 1NDI(IAL FORCEI.:, 4sI D.FkAMMIN1NG \VIN5., 13
No.\t I it I ./ I. anid n-sitig I ,' Is (lo.,igtnals tihe pithill iu i thai ik, tI, lift voe'llicihil fill' a pitc'hlng %%i g Iljia I h,, ilt 11tt
titolt Wi I t It sil l t I I It l Io iiv I a It ig etge of I " li witlg. of th ie lift w il t I g ltsii t sl twit'ivIvili t wf at 'inkitls 1119
(p illtti I ilidslhatv thI wit g i. pi t1lihinsg alsoiit arid f ill sie it iit re
-(/ Itieasltts'i aibout tiilt wiig leadilg edgeI. Helect.,2f---( -- [, ,112. T: ,-,22 + I t< 1 " < 1
<~~ 11(
+LI - - -
I~ Af},t,, Al-l." A t ,, A I I,'t"`,1-. ." <. 2 f
I<F , (! >-I >15.1 . I- I!-;Iv,'-- t l- I j.i]' . ". l - -- &t" -- - l -I • I
"rl.he. ,XJls,.sios.s agrit. withi those giv-eli, by IMiles ii refer- A further a ppliea.tioin of equation 1(1:ua provides tie pitcih-e1555 2. ing-monterit coefficient for a pitc.hing flat reclaInngl]ar wing.
"T'h aboyv ,r'esutis (,all be used to detmotnstrate t Itseful- TIils. With I o-.j I . it . eqluation (3"a givestess of equlation (:38a). Taking j n =.q=t. 1= 1 its that
iqia i Is a__s k•F1
or. for the pressent case. .f00- 1h(. fi - fl: )atnil so
which represents the equality 2E =\ -a 0 -11
" , (' ---+., +C'.~ From equtation (B21 ill Appendlix B it is foutd that.f` F 15 4 tl I
20__ (1-+ft,3)-1to- [4-M(Af2+3)toeIJ ()<t.< .1
_u",, I,"
4 f I [1 -Aft os-' lt°- I I M o-(Mf)
=+:-41 T 3 ; to> l
I lo+(l+24 t02 tlf)2 +S 5 f(.I 3 )t2+(.l- 1, k3 /54] ,--ýf } )3(, < too <
C'ombining, We finld
2= 2 f 4-MAtný' t,, }j~ 31<toIf M 3 --121 [- t+M +-:t.I; o<t,< 1 +
9~~~ ý I [2+AIj(o _ Al., 1 1ift,,-(AP-t-2)to:11 -0 Aft
= .: t , .
40,1 ] <,,,
14 i•l.'POr'T 1230-NATIONAL ADVISlORY (OMMrITT-E FOR AERONAI'TICt
Aniot her relat4ion aluioiig tilt glertioal izei ini Iieiai forceis jJ'iv,' '5 jiall be derived IbV IllaloDS of the reciprocitv relatihsgivon i s -- "" 'v' I -j "- 3I J 0 I !- ;
refervl ie 5. 'rIll t details of th h derivation ar, give e ill 144)Applendix ( akind t11re rvsults t
The il'ls). snl oil the, left ca) Iw )(. h . rh,,,
4(44a4t 044 (39) 4441 b)e tlset i ilk I o avtlSv (til'. als a Illeil.s
for checkinig the, internal vonsistin,.y utf a set of ahhllulteld - , P) 1)generalized inlicial forces, kind tilt, other, as it nmeals ftor M r
expressing at gi'll force' ill P 4 4l44S (if a set of ollhlers.
Conisidler, lus t44i~ll 44llpli- (f Ow former4'4 use'. th case fill. tiip,
whic'h / j 4. Then ..... p
E1111111,4 ill'- 4'4 icivil 4 of r,
FrloI eq( ation t3ll 7) 7 we (tia1 expr'ess this r'elaltion4 ill lentls of P I P ) ( p
the' cahc'uhtted iiiat l 4%"4 till4s ,ar 'r
(l) equt-ion (40) bivtoll
.• .= \#I L ( -
. ..•, /•2} F', ,
('ONC(LUDING REMARKS
If 44)4W It- 1. 3 the following relation rslilts A nlc(thlolh is frosri.| l for VealuIt inig thil gvllerailized forces
O44 44 rittla l.iltllr winig hlyitig n lt a 54l)4'l'4 llit s)'e'(IS 1144 Ilhavilng, ,' ,,'') .I* -- ', I 02' 00-- + til jsl( 't iti sUCIu h that I. I > I. 1lt'h ge 4r'a4lized I 'oord li-
Ilat."s use.d to 414hlne tile wing's Iwhavior li't( tlt- ahllplitilhRs3 •"())-- 'P'-)2) + 2- ) of iownIaslh ,ist rihtliuions4xl)rssi 4 ilsof polyllomial-
"oz) V3. j( ill .r and 4 y, th i(Ihord1(ise a44d spainwise direetions. respc4.-t ivevl.
%%hih prov ides a useful chu eek o(n the eomnpu4tted (qualn4tities. Nril Iesults 44're prit 4nh' ill talde I for gen(ralizeNext It us solve eq nattion (39) for a given forcei. Perform N results ire pent il l i able I to ofn4ral444dtill sul ovraionindlieiel forces ol it wing having all aspec(t ratio (of 4 kind
flying at ia Mitah Iinumer vequal to 1.1 and 1.2; the piolylnomial
, ,J.) Co0 'VIi''gi beinI 0 < 1< I kind 0 _< ,, < 5. whe, 'i. ,1' - , .j (- i)' •AMES AEo RONAITI(A.I LA..\iIIATO)RIf
on both sides of equation (39). 4444d1 r'eiver'se' the order oIf N.'.TIONAL AnDVISORY COMMITTEE FOR AEIRONA.'T[I'.st4InimItiOll t)h 11 l left side. There r'esullts5 , MlIOIFEETT FnIIu), (CALIF., 1i111 30, 195:;
APPENDIX A
EXPRESSIONS FOR THE POTENTIAL -- I l-. f '. )
,II order1 to write tlih expressions for tlh pItential in all (t- t" - (a-- xg):
regions Slown ink figure 4. it is stitciveint to derive iik uiet-ailonl, that for reg, ion, V . H avi,,ng ,arried ,ot ,his a ,il vsis,. It"• .. ." Y" "
Olie Can deteril l ine the expriessioi•s for pot nltial ilk o the r r . \ t-- t L) ; ' - .
region-s without ditiltuy. (A2)Considehr, theref.ore, eqIlat0io1 (13ll an let a and r apply
to region 'i,. First,, it is i ecessary to d. terlmine the Iteel -
tials It', anii I ill the t.xt Space. From eqtuiation (I1). in .\'l U-. ( y,, -j,- )ioltjjullctionl willt ligure 7. it is founid that I 0
.. ' -+• ':-" I'- .- r,'+,-* wt~ .lit,.,)4 1 \W itlh the %ailue' of 11' given ill equat iont , l tI and (.lA21rfo 11 it is IMssildt. 111)% to sol~ equ(Iiatin ll ibl foi '. figriure giviig
- --- ' the requlire.d data in fill' 4•.! ilihe. Thit, if /h" (---4(Al) (y -y•',
Y. , ,fl.. .y[ ,/ ' , ,l: _ ' - + ,• ,,, , -,•
"lrJ•It B/- itt- IVA'-'J• r•-
(1',,1,£ ....- '-'V + J '!h .... J ,,Y 1,,,, •
Now applY Ille operation of equation (7) andi the potential cr, is givel bI y
di' , , , _ " , r32-r'g ,+-•" .... , ,,1
f - % - -,t
IF + •f, (Y , • 4y t
whee' R 12
= -(- Y)2 and the bars oil the integrals signify (AI), we can writethat the finite part of tlhe integral is to be taken in tile snsdefined' in reference 10 and that tile order of integration I [ , ''-"
cannot, in general, be reversed.2 J-, f.Y,-- IY,2 (!/-/) 2J 3 2
J- 1 -
For convenience set fl- . :+i: w'(t,.-.[,,)dl, 0
V,. -t* . (A5)III order to simplify this expression, the order of these in-
where I. is the nth integral group on the right-hand side of tegrals will be rearranged so the itintegration with respect to
equation (A4). k, can be carried out first. Th'lli technique of changing thle(Consider the first of these integral sets. Using equation order of repeated integrals with strong singularities set forth•
For til subsequent analysis tho old. the definition of the finite part given in referenc, 10 ilt reference 10 will be used here. (Consider tile change ofis e•sential. This definition dilffers from that given by Hadamard when it applies to multiple order ill the tZil planit. Pretend for thl mlolmlent, that tit(-integrals.
! Since the order of Integration plays sn Important role lit the following development, t1 integration has b)eeni carried out. Then the highest orderitegratlon first with respect torand then with respect toV will he denoted fdg fdzf(r, v) singularity (since w. is hounded) in the kt, r, plane has thlwhile Integration first with respect to v and then with respect to r will be denoted fdr fdv
4r. Y). When the notation ffth, M) dydx is used, the order of lntegraltin is iimmaterial. order 3/2 which is weak in tih sense, that no residual ocutirs
15
O
I t itEPor L23t)--N\ ' l'llN.\t AD% 1.4 li( 4I ) % I N I• T' -,•-E H A•1 .l} toIN AlI '1'4
01h14ll I h 'lltli44' t4, llte lalit l i .,� r',vilrst-tl. Tilt, htllJ to ligurt 15. it4 this 'c u ii uiliiiieii lngidariti. .,xit-, ill I lit-
tlll II 15" silt ti t, t ihet a oa f inltt'•graltioll, so inllhldi'titel. itll t'ilile ,f I Ilie ýili llkIlt % l ti' ofI lit, li•i t-lrnind: iln Iv
_while • / y lt I, y t lie d I ilIu•-tIgt' (if
J, . YIi) -,.Y mlull s t I ll ti k e ii 4 o f t li t, ex i hst llen ot f it rl j i h ia l it4 'l ' c c l' ft' . i l l .
'- + 4.....' +r .- l~tj!/ ,i~t• TIlis rTh i,'s r illul is d ftihlied as lit l iiFt- t ,'e hctAlt , till ht t11 ou \(I /iJ2"-(.r--,+"tlinte' l rtdll titke l il ,dilfferent orlderl', ove'r at vtalishillgl% sinall
ltegilon surlrtIluldilnI the inhtire'tit silivilhlllit 1 Ilthe dttliii
"T'o c.haulg ortler ill the t. f, plante. ,tllshllt the b14lloll1 of lr'ionl i Ii, li4ttom of tiglit' 15 1. "liIt', r.sidiil It, is thin.
J 1*(, £4 .l/ti,,.Adtijt4~
0Cwhere v r,, -If x y- y, )-. The settolld integral vunish,'
(ee ref. 0Ill, and. p)as:sin~g to tilt, limit -0 ill tile first imtegalthere result.
2,• - r,, --2
where thilt' sqluare' i)ra'kets itagaill lilt'all thIt tile retarded 0valute is to l take., iThus, the iltegral 11 allr ,be redtucelt d ti t
C, -Y. -Y,- In tile same way, tilt, integral l2 can le reduiedl, HIM4
'X, l --2 J ,- I, - r,,
which is recognized as Kirchhoff's formula, eqtiation (3),
with all acoustic plan formn lholunded by tihe vircle
+,(.t-.r,)±(-;-(yt-- ! 2 0+=t
Trhe reduction jof the iltegrals I:, 1. I1. I, and l, is quiet
similar, leading to the sum
I I -I I
(A7)Examination of till' limits onl these illnt•grailwsshows their
total area of integralton is that slhown ill figure II. Butthis area 'orresponds exactly to the aioustic plan form S.for a point in region 1",,! Hetnce, dlelnotting tile combination
of terti+s ill equation (A7 ) I)y c"' we (-till write sinllj)iy
| '(' I" d ,•,, , (A.s)JJr r,,
It now remains to valctilate till, integrals I- through It.' l)esigunating their total effect oil ti' plotntial, n ' one (ollt'1ll
readily show (since no0 inherent singuliarities irise ill thilteselirt-It,r lS.--Areas of integ~ration us+ed] inl aualyý,i:. (, t.l,' halt
. . . . . . . .m ... .. .. .. .
G;ENEIRALIZED INDII'IAL Ittll~tIN Dt..I.tiXitNI` \IINI ,2- IT
J .. . i. . 1, 1." + I 4!y, ' Y,) It,
livh re 1:!- ' ~- -.I'[ " I~q- !,•: xo% le aln~d oik] lilt a'ousli,, hlJla f,.,Iln S,.•i "%4n111' fit le~ re.gioli
-!y 11j ' t ,)df In tile clt4 0if .~ till part1 If tile poteniall l ll-ille i"I 1ni' of tlt~,s'ih SK v d 'i V i i" tf I,.V i'Aing. vy•.io \ AI II h$.a[I(
('),,.,1 1 2 i..nt-- tl ri iq II)/L ge..neralized and written
4 i 4 11 " I',(J * .M tI.!h )dt I .. ..
In1 Ilerlls of this e.qirlessioln, vy~q lition (A91 c|all hez wrlittenI
ililpl,V whr flie t h iintegrainids art, defined ill ever) cast by eqailaitinI, jJ ('(p" . lb did11 (A tA10) and i ,i ,tie "refliected" acoiuistic plan fOM S.
Changes wit i thie region. The regioill .S', is ahvavs bhut idhtIiy portiiions itf tilt' "refletehd" eih' ;
\Vli'e, tlin arllea (S,) -a is illusit'rated in figuire 12.hIi order to give expressions for tlie potential in everv
region of th- % ing tihown iii figure 4, one ear show that it is and tilei "reflectited" clipsetiiiiMY Iiissa'V to vary the areas over which the f lt-ouble iilntegratn tion in eilatjlhtslls (AS) and (All) are carried out. ( - + (Y-, x,Thiis is ev videtnt in conntilion with the source portion •i. .i/J ' *# '
for ~ii ever,% c aser ,rf w.l 2) Figure if; shows sketches of both `;. and S. for all regions in
"[U.Ily, (Ad2) figure 4. Trie absence of a sketch indicates that the corre-sponding integral does niot exist for that region.
APPENDIX B
THE GENERALIZED INDICIAL FORCES
THE LOADING COEFFlICIENT and
li order to determine total forces acting oil the wing, it istirst netessary to obtain expressions for the loading coefficient. Y N 4VYI U,(O,y) +t-Ap q,,. According to the linear theory ot t+R ) r/ + +)r
Ap 4 (3
so it is necessarv to differentiate each of the expressions for )+ d, 4Mtlpiitentiial. As an example, consider, as in Appendix A, just 0-
rigion IV,, of figure 4. The loading coefficient will be dividedinto two parts .p'"/q,, and Apl2 /qo to correspond to the po- Notice that if w diloes uot depend on (x, -+ .11ti, tilt iintegral
titntials p 1 and r . Thus, using equation (All) term in equation (B33) vanishes, whihl if it does, tlien the
integrated term is zero. Next, for x ý>0.(..p '2 4. j+ f-• ' ."+,,t-(,+VI. (' _
ii dq. dr, 4q-o 21 - , - oM i z,1 ,ly, '( xiuuI) ' Il-
(r2_j,,,) % r "I--r ,"I
04 of 0('t f ,!,
sinicte the derivative passes the x,.y, integrationi withiout (114)effect. Referrinig to equlationi (Alt)) for tile funictioni C(x '(i i/i)ive n~ext find its derivative with respect to t. Write i= t-t. [~In litiý ealst, hot il termlls exist liiiless wr, is not. a ftiiitt inn (of
Owni for r<O (+.-•+ in w ii ci case filie iniiegral van,-isiies.stillit io of e tia tio s (1:) Iai ( 13-4) inito e tiat io
o? 1 w ill nlow 2 -d a ex'cpressio n for lie lttinh g c teffi ient
(I (It. ef, rrespoidinig 'to t ile i in i hiithint of flon sitit edgeW+liXI{|idis ervaiv wil os~ct o . W rter= -- ; i i•l ttSCloi li'ln e it llh's t, s t~ll afil~iliill0i
REPORT 1230---XATI)XAIL AD% IWHY twitlirl1rIh FOR .AERONAUTICS
- --1 ------------
....- -. .....
I7 , I ,4
!7
½'
F*i;i RF It; -S-kete'lie. of arveas o)f fittegration, S,N and •N., for all rv'•jons FI,;iTRE 16 ('ontirowdl
Wl fi1•mr 4.
( " 4 '', V-1° ",,,"+ ":" " ' 1'' ''' 1" 1 f ,,, ."' q' f ' ... 1. ' -" N'4 y" .r," _"1'" Y1,"11/-I .. X M",- 1 +- .
N~ 4I~ I. II) 1v
yr .
The vxplitcit form of rit- givon Iy er quat ion (2), h fr lrei Th portion of thE Ing -oi'ivit voir nin int to
in.s'rte'd and it i.s jjsstunm j that I>_ I. ca. n be fou.d readily and is
-r, r .I ' i* ) "1.o-- .. I. ",, ,' '' ( U .~- t , -: )--( -,j-- -:,)) i, ,
J i1J •* t-r"l J I+ '•-•°Y r -,% P, J (y- -,,)t- 1 f--
2, 2), + - (•1 1") - yo I, •
(, ,, ... (,.p €•(,_,,' (Y")[ ± • 2 (~1)!t "'C " •tlnt" 1)
f~i 1'n'I~
I'ii i
i(.1fj4 1-! Ill j-, -.1i(t-r,,)I' I
fii+ /Y drA!t-0)'- -
Y1 '/y
GENERHALIZE') INDI)I&IAL 1,01 VS~ L. N D)I-JOH \l i N'. I iNGC 190
Region S So fit- - i ,IJ" P" ,Iy I•I%i v of lit-( lo d n covilt, w l Ip' qJ. a t oll
......... a t of ri t ipp n iv ~ l jfr. i
ilh l .lil - hl'l [ %%O ll h I x,.l '.q il'L S .. t l ill ' l tI h
it i__ _ __.d th at Lh, , ti,,. d2' 4,, ,. ' , , , li,,, i,, , i. ,, 1112
iif all bca hat. yen % fli : i i ruhi er f'lo /" Trill '
tii. iYepo en ial rI in. i' l i'ilar, giiiri oliri•ivi l. a.llI\\ , ollt\f,-ota
rig Ac ~r'rii~h. lI~lI)'hls. 'I'i e N Ill~i~e . ith ilj•'i ll .1n~, I llrl rI'- ei'l~e rI-, . lh-l ,+llhiltI Ni tnd i2m II
K i
II hu h1l)(Ii'litl thl t1h'ilfi+nttr nothiihitht ),)'h45J)l',.l•.t~g rI'lVrII-,ril i)1t;,IrI
V I iiliirI io tlll tl', \1tl 17C.ri,,t, l l IiiI.i'iluf.llillli.Ih,
"ei '" ifif - r ol if r Ile of total hilnt liTictrn i 12\i' iil II it in l h i * ht, ]p1ir l illli I i d if l ". lilt. \ , p t (P'klilllil. "¼ I l i' . lh ' \\" A ) \•r.
vlvin I va Il I is t n1011 i f I) gI 'I sj f(Ifi 1) I~ (f I Y pr 4 4 ( I I be r), Ill 2 1 1 lilt(
litirlt )11 8Io i ts fo im% 1141 m lop iIIWI Ow ci vnt ion 111 21 t1hu" if,,a T* i ll -l .s7)11 '
o 11! / .) .. .... ..7O q.cq,2,1 2~q, 2J 1,A,,T p~ 11
) 1' ~~i•ii• 2 7
2j•'.v 3,I ," 1 2;, 2 il' . 1' 3,l 29 .. .1 1d 1
2 2, 21 2I: I r
Ill va cll,:ii lL hil t- ,x ',llerir llizied ildilcSial force lll', + bv n i ill" ofp N 1 2 .- 12. , , 2
(\\ 1ill iollh ( 6,l i t h' ] ils b -l'h shil lh ,I! tl iih'il h 40111 t ihe , a llm .V 2;l ) •
fi\lle l',+ <'l take fo i' ll,\,-, in e . riii s Ihi , s <'tl fiiil I ii 1p+ ++1)h''UiI lln ih s x h l "I'
•,l~ l~l,-.)•l iil h i il hl\ ii'lli.h i i-• i'i lti itl i ll'l lll i'l lm i] h
I~t .l ll•! '; ]l!/ #•] I. i< 7 ; ]• , ' > -
0
2t 1 KOH~.4T4 I 30--VA11VINLI A14\ 4ý.4441'4 4 \I I l [. Hill AER"N\AI1 Ill
1, 44 I -3 1
1 2I - i m
1 1 - 1 53 332
I 2)t- 7,I++ +! -I k : I 42 ;• + - , 'SI4"425 30 - 5. : 35
10 312 ii 241 I 12I 1 532" 2.11 .
2 J+ -45-1 1
t .4 IS 2, ,4 4445 4 ,,v B ( ) , ,.,
_~ p.
44 ,l I¢p, q/ I' ti4e ]4e4 1 fulletil4 n. I K.4)/r [1?l , v - 2 *)1
1fIp,.) I: J.1, H( .i)T W r "2 21 2+ 2 2
2, 11' -'" OI a' [il) -I<.r(<_ .-. i.cos -- -•'-
'I'll f44,[4a ..,.,1 ]['p) q) lI'(p ,q) ,,I4 P",.
Th,, llicio'lh Jtq, to hills beenl valclh'hihd for to t aken
(1. I. 2. :;. 4. 5. B4'•'a444 ,' of till. I)n'4o rtyN. I Itri it i, only (iv) .11t < j-< t: R./. 0 ( ' ) 4
I4.'-."•'n'y IV vi\'. 1 trial4 .14lar al44rr1y. %%hI 'h apl'4rI ill t414aho,,. talble . K ,.(,,)
No%%' c'onsidher tilt- func'tions lx',,t) atikil K.Wipi). deh li.< ftvi i iarIn a iiiNtlrK v,1-1kI44Illi4 4li4 ('B.12 Ii is ,4oive'll ie'l , fll 'r o4 llVll)ll, l+Olll| pll'- 0
4o 4 A, t4 •' '1 x p ' s s t4 i l e ( iJ'Vf e ll I o 14 14.1 i n c o' lel i ih t e ' blel ii l .0'l. '+1J
4444,' , 1o , 1i d .i4,4d z'4
1 11 1 < 1.... .241
B , , qp. ,l 2 2 4
111,, 5ki -t .'< -#l i.i.l' '" ' " i,(. )_ \' ' / 0_Si"i4. ,, t 1 ._• ),.-.11. +. ,,,p ,
%% ilhouil li .suibscrilp . the 'omln l ehtl integirli sll~il.I lli. (x) ./ < r ,K ,ll .[!,' 1 (• 2'~ir~l', I . N t' tal i's Iii th. Illii is. ]v)i.11ri<lx,<it.
- d~~1'A4 v ,2'
lil elimiitionl (B151h). x" eI,<tlals 0. [I it 'ttsl to e'xerc'ise'.111in 11rl, v%%]-heni ii erp tin 44 (4• , 4) and] 4" its', 1 1 bet The ._lrn ralized h li' l fo.' , F ý", call no%% I.,
f'unctions+ because of thcii ppter limit. TIim . since,
4;4" "f'...II4/'/4"44I 4 , I \14-4444 2,1....44' -'4'""4I ? p tri 4'
441,,44 ,s i"t4). 4114. 1,i. (iv) , i ... . I4 " 1/ " -0
, -.44 •vl th e fol4J1. i ,I ;A igit' I 2 m
(i) 1, IlPt .. 4 c '(o 4 . 14 1 l 444 I
S. ... ),I t. R .P . ]d .. .....Bl 17) ; ++ • ; •
3l,. OilAlL.i IN-ý [IlA II In{ I. IN I)A)AI( i~ I~i 21T "
R. II
il v V 1120) ii
It is 333j,. 3ui3In h,) express these, foiiti', ili i 7311115 Of dilnI'll- "+ - "/•
• i•iiI 3e-. ,it ll lit I 3s. 'l'I1,4 settWig. -
h "2t e • l " v cl)b u ri.- I*I
2,~ ~ ~ ~ ~ 1" 2m1/, 2
1I1c 4o' f ,"" c 111 l t. ',•. I 'll, ,I ,, (11 be Il v l Ip i ,he. il t 114 .' IIII-Y oI - "Jfm l 111
-) If,i, ,ii ll .l', .i [ -ol I .il I
""-1, 1+ln fi , • i t, .%% - ev fot lig in 17 O l-lv'i
for the Ilp cl-linlJ of .f t-to +illiv-l .t ] . .... "-: if wv) (14 2f0)o
: 1 1c h I, , 1 _
B.I 11, ,
Iv.,' r I + ) l3l' •l3'r lig Il)ll. "rhi3' t3'Iltn) h t' ,<tnl~ li'3 slie . il.'nhr~,i ('11331l1 13) Y) -. h p
-331 1113.h -f)' IIII31lMim it 3f 1r t I 1v n ierlyl 1hi w'l 3;.t i f• lh I, r .ii ' v 2 >, u ....
il h ' l'll l~ il h e .. ,. p ll ll . 'l l..; h~ frs t (, illP rll ]l~ v +1{ I ' .. ?./t+ t ..il ,+ f+!0
I,,v•. I~v .'l h•• upo Ih . Ivht il~l Il~l+.',Vl lh , c ord",• :} v -2 .,
lel+ th lli thp +•illl. X4' 't'4 fll~ll li un'IT h~llrq~l~r il0
APPENDIX C
D)ERIVATION OF RECIPROCITY RELATIONS~
q.. 1 / / / " ' ( '2
IiIIa 1 1li I.., . i w.
I ~ ~ ~ ~ ~ ~ ~ ~ i it I] IvI It1 Illti i aIol
t. lie i I,. IIII-,III -IIII r f v o w I w- liii' I., II I I g18
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Illm n'- illOw .11)1)ý tv 111,cli l il llt- Zjlw Im lin -I hr0
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PC ! II IT i i t I ý I i i 1 2 1 1 1 \i
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2..2 2' 1' 1' 2'
tin~~~ 2 - i - -
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I ~ ~ ~ ~ ~ ~ ~ 7 mi t2i 17t .II1 it 23l I;t I'+ .•1,?i • I ll p ,I l' 1;,•; : 1 l II I ,I
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1•tq " 101• 7* 173 1 :.,• 32l 1 3 2 1 1 .• .13 I,"9 1 k 17 . ,,;' 13 1m .1 - I 1 1 14 2"'1 i't ,l IN I . .2 I A 1
00 No•I ,• l~ I IV _l ;I _N 3 ¥ IIA .I Qi II 1, .41;5•:• IU • I+ .•I ,I iJ4 1.~ "
2 77I 1 1 l: 1? t. 1.. .1, ;. 1 •• 7: ,1•0 III til ;q ) • 1 2 ", l.•l I ll I" !.,/4 +I _N, ;.1• 1. 17 + 11,',.
It~(. 16 It: "21 21, 1 1.1 , 1
11 ;' 21 ,le I 3l :l2 71* li; 4l' I 1 14i¢ 2 44 16 ! +1: : i N: • -I 14-• ;ll 1 1! ," T+ Ilk II 4"1 117 It- .,
I1 1l, &72 I,-'! ,V :141 _I %11 V. 4, I 1 .51,1 11 -4 N :11 . 1g 17 11; "3l•19 2 Z; I7 1 + ,,1- IIY .11 1
14 t:,i•~l t1 2:• I f'lI VV llq w11 ; a tl7 1 2 ~~ 1 a'4 I • .+,2 :, .I ,s1Z2 1,9I illIT -I•4 l-, lP* i t .I Vll ýio41 t•I 2 1 ýNi . 0 47 .13 . I 'l 3 I 11+- 74 ýN.•l 4 ý -1.+ 1..] I -' ;
", ,l•I • :~l I•'w" 2 •• ;I , 73 I Ill? tt 1177 0, M I., 1, . 341 LV, 11 *I 1 7 1,;+ 9 1;.,; 12, 4' l 1 2%1 !411• IT V: 2T4 , 2!2I, 01 1. 9 13 it I .14 il 1.; + 1 7, .' o I2 I . Y * Ii I• 1V 10 711 -:1II 't • 1 ,7 :411, I
2. "'2 1 1 A. ;It it l+ L io • 711 w, ! -1 +1 I ; 6~ t 44 VLf ]I ii*• .9.11 1+ It,• III -ill 7% 1i '-1' 7I -2-' I'M2
:I+ 2 I'7 itI,T 1I 17 19 !1:;" +3 1 4•' IN- :1i+ _,Il I ANt :,4 I; :i 1' +13 4t 23 , ýi J."l ; .' ,1. 31 , .C..-1• e i i1 1 l t. " .MI ; ,t: ' s2: 1 ,P III) It7 It _1 ýCT 'I I~ '. T."+ II? II .7 .1 +21 + I I
.Y2 Z 13 (17 1: ,1 2. i v.4 : I. t•J •l IT I I•Z W i III I[ i =t l 1' t I ;' [it..I 171W, .4I n,• I• M1P ?i+1 •I + • il,?
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• • 43 I:2, 4iI< 9 42! 3 + I 41 T I 12 I 1 4 6 16 2.~ ;4 .1: LM•' 19 6II' i• 16 424 :+1 " I ,1-• 1,3 ýtl 17I I, "1 2il I 1 2 :l: 1 4 t•2 3 1 it4+ 21 It I 1 141• 2I1 *Z, A 1,* :,i S3i9 •l I~r 1i 2 31 l• 6I• At.1 1!R ?4
.I I•; I••ll ' 1 I I 3 :1. i +5 111! 7 .) 1 I -1 I1 102 I 2 4 S:r 3 = I 140 7 13 ; IA T II 2 Si 31 :1 77 jI In .I .-! 7•
14 ;-: 42 r,1 9 2, T 1: 1 1•' :1111 , 7 71Ni I ) 1 Ill 1.1 KI: 2i :0) 71 9Mi I 'I d 13 1' "1 1* 149. 2• !'A ;1721 111,• • :II{ •4J 1.: t !I4 ?1 1 ; 117. 1 • ,M 2 176 M 2, I; 15A ' IAM 4'. 1 2 5 241 :1 , ~ ; -711 7 117 !:3 52 1 Il JI 31 , ;1 ,AillI 1;1 37. 11i~ II zi•
I o"+ • ?,: 1l 1152 1' -ý i ,• 1 -2 I 4X i2 l4.- ; ? i: . n1 711! mm Z 1:1 i97 17124i7 I 1!, 13 :,14 2• 1- l.12 441 2 :2~ 21i LA 41:
GE-rNIERIIALIZE;D INDI(IAL it4111E' ()N 141'.f4414N N4; VINGS 2
"TAlIsL F I. N.\lLll.'S (IF A NJALHi.IZ UI) [\])[('I II, F l( -L.S. 1-' t , i~lmi
'41 .I . I. Il
'44 1.- 1, < I
4224`2'1I 'I -' 424 At *, ' 4 I ' ' I. I . 4 . 4 .' 4 1' '. 4h, 7 7il. 1 '4 ',-44 4 I ! ' A
i t 7' 44' p. "212 14" 7 . ' . 44'
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I ,'I W''44 1 7. " .4 I T 7 1 '2'4 -4 7.,..44, '1 4 i 4 ' ''414, I"' " ',444 '4>4' it ,'""7, 4' I "I'l ''' ' , I 1 4. 1 77 t 12 .1 4 , 2 I 11 i 211 "
77..; i ::'21 ." N ' I. 4 m. I Lip. 2i l -P .' l 3 !t 2 J,17' 4 .9'. 21 4"V. . 1 7 . I A I' At V". ,414• 4 4 :.'4'I, Ii ' '.. r IT
t•1] I I 7 I iT .1 '7 Il. 11 I 11 1,i jll •1" ii 1, 1., li I ;. il l l , F 74I 444 9% 1 7 1, 0
444 422 1 5' ' 44 Ii I
'.7 .9 4' -4, .4
. "14 7 i 19:44,; I I 1 4,4.' 22 444P44.14 4, .I' 'l I4I
4,4%''5'• I Il• V , 4 14 I t44' 44','.. II " '4''I - AII q1 I444%'. .. 4 '' .. . .... i' i, '.. ...... .' . .. 4 . . ... 44 , .4 . ..4 ..42 2 ' '4 . "4 4.4.. .'.. .'4_-... 444'
1 . 4 44 4 44 4,4 7.7 14 .74. 11 140 1i 14 4' 4'' 447, ''' ' ", 44-4 4'' 14. 14' A' '4 7" 1u w I1%4 ''4 44lt I ; I .7 47,' '.44 V.
41 14 'M' 4 P4 4 4 Ii4 I ' 4A i !I 2 4'' 44' -* I4' T"j -l It .°9 ; , 12 A I' A [',2 I. ill I I ' .. t 4Al I 14.4 ' 4 27 444.4 'A0 444 4 7 - 14 454924.47 , 41 44 .4 4ll n2 A 'I.1 W,.4 12 7 *2 4 !4 -!t,4 4 2'.' ' 4 11' 2 7' 411'-
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" iII.i: I. V.A\iI, . I)1F' (;I-ALIlII.ZFI INI)I('IAI. PIl 'l , t'; (' 1i411.1..e, / 4: II4 .11 1.I
7,,* 1 :o I I 4i i ;"1 1 11 2 A .ll , I4I , .1
:I 4. 4.l 3: 1 44.4 4444 44 4447 l 4 1 4444 4 7•, 1; I 4'. l 47t .I" ; * ' 4444 ' .7 I i4' Il 1 1"W 7 177- u . .
42 .47..'. • . 44 4 4.`4, 7 i 1 44 1 1 414 'I 1-.2 l44 ' ' :4 17, it,74 4 974 4 1 All41.4 14.7 1-0 1. 7 7 :.77 "4424 .40 .44.4 4 V, 11 412 1. 91 3,;C4 44.1.45 4724 10 t r 414M44i44 .7 I4 .7 A 1 7 1 1 31 , trI% 44 ip, 4 5 ... .. 44 34 14N 1 31.2 4t4 . .2 7 .. 444 ... .....12 'll I '- M 17 4 it 7 4442 . 42 ;U 67 0 "4104' I N 1i .4 -4. 'W.4 72441 11.4 444 21 2441 A 1464 it4147 4:1.4 3! 49-7 .1 44:7 4;:7 '444 "1 '7 Xf4.44',4W43' :G"' 344 4., 7 4245 A- % 1 24 W A 2 A . 4, I4I .4 ti.'I I44 5;411 IMl 1. . -. 7 4l' P, 4'! 3447 " .5 :
t4 l ; W . 44 7114 4 - l 4 74 2. I4 4 ; 1W4 4 . 474 t 444;44 11 774 '4.7 %1 44 44 4 4. 74 7 1 4. .'4 . .. 3: UI 17 i ' i :21 Il;4 1 1 " 41 Aýl74 44. 7 :1 4372 :4i44 i2'67 2147 44444 4.4.t. 70l 1 !14 7I :V'*2 ;1 7 ... . . 1 21 1'% -4 M :0 I 2L, .W 147 7 :4l I 1372 W'% . 74.. ... k c4A 752 Q 1 2 1 - 4 11 U 434 . .1 .I I1 '.4 11. 62 ''4 444 W . 3 52 5.4144 1,; 4 Ti.244 :4 ! 4 4 I2 $ ! I 1 I. I, 1 10 'Ni 44i III% I41. 44 7Ii M4V4 I ! 1.1- 1 i': 2 ' I4 9 4 11 4 4274 '4 , ..47 11 W4 .... 4 17-7 ..... i --7 9 ,74 4 4• '1 "1 Ii . U117A - . .. 7 , 4,+1 ' f... 4 ,, . 7 ,.42;,7 42.4' '4,117 4,.,,74 44,7, I., .. ....1 7 .7 , 4 . I. '- t'•• I !,w-I ý,11, Il It$, :,01t 9' 1 G 67 I':t 1 i 711 li ll .j 1, `11 i 21", 92 1 ,. 311 1; '12• 11. -11 1, v, 3• 7 ;: 713 - 1 2!, 1, ;
Af Z'W 6l • 1 ,.i I% 1~ ;1-l ilv ,7 • 171• 1%1 13~ 1% ?rt• .• W? l.-; i11 7a~ 16•i 2 11l 13 ay. 1. A{ A t •13 J.-I -1 2114I •lI . : fi'. l mi wI•, 17A -.• ;1,!' .l ; " 17 1 YIN 1 I -1%i 2),% 1 .'1 1t7 I.I 711 1 31i 1" ,; I•t ,'" -11L••1 , l -ý ;7 1 ;7 17t 7 1 3Ai •I t 4 # -,3 :1: A l An I 11 , 7 1 i :0- I I ", 4 33 t i4 t t 7 ; 7 7 1 1 li • ;11 7" 7v. l7! 1 %' i
.4 ,;7 : 'i7 4 174 47 : '. 1 11 '1 1.,7 7 417 4 ,71 544N7 W,7 , 7 -. 4 4 Al. 254.. 41 -,1 1 17 "1 '017 4i4.7 ;7 14 ;y. 1 ',Ai 4I W 1 67 17 7, 2,;4l 753 .474 M t 4.41';4 17341 15 475 44 ' .4 Atl 444 4I44; i7 . , ,,, ..... ... 4.,,47
47 7, il444 .4.4'X .445 t 171:, 47 N- ll ,4 54247, 't... 3,; 4444: 4o .t .I 4 - 1 7 4• 4 :41 574 *'',, t05 A 44 7 177. 7, : i.4 :'! %if; !l. I4 I7!1 477 444 4 i !, I744 C. 4'4 A4 N1
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'11 3J 1;7 711, I, 1 ;177 2 47.44 2' .4i 5. 27 4 141 4 2A 4 ! 2 'Il I"4 7 i4 4 7'. '17 11 24 1; 3 I4' I It.I" 4T4 4 4-% 441 4• i 21 774 1 17 1 :1 1 Ill I t 2 T . 147 1:4 11 :9 1 3 73'; 4 7 '17 1.' 1." ' "1.- 7 2.14 2 A 4l' ,• 4 ,1'
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