The Aileron Influence on Wing Flutter Citation Kybal, Dalimil (1940) The Aileron Influence on Wing Flutter. Master's thesis, California Institute of Technology. doi:10.7907/46zn-zn97. https://resolver.caltech.edu/CaltechTHESIS:08212024-205858806 Abstract No abstract. Item Type: Thesis (Master's thesis) Subject Keywords: (Aeronautics) Degree Grantor: California Institute of Technology Division: Engineering and Applied Science Major Option: Aeronautics Thesis Availability: Public (worldwide access) Research Advisor(s): von Kármán, Theodore Thesis Committee: Unknown, Unknown Defense Date: 1 January 1940 Record Number: CaltechTHESIS:08212024-205858806 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:08212024-205858806 DOI: 10.7907/46zn-zn97 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 16663 Collection: CaltechTHESIS Deposited By: Benjamin Perez Deposited On: 22 Aug 2024 19:18 Last Modified: 18 Mar 2025 23:58 Thesis Files PDF - Final Version See Usage Policy. 17MB Repository Staff Only: item control page

Thesi:., by :.1uli1til r.ybal In lurtial Pulfillroent of tl.3 Requireffients for the uegree of t aJter of Science Culiforn16 Institute of 1echnclogy ?aJa en&, Ca if~rnla - 1 - Thdt the bi~~ron ha~~ 1~rge infl~enc~· on win~ 5,& flutter ,,as fou.nu as ..;i.lr ly as l.J~O wten Fra.2ur and Duncan ( ; ublis.hed Ailerons". ~ 11 Llst or l)eslgn Jf.;Co,,.. onu.ations rl!lative to Since thes1.:. ltecv,,,,r 8UuatL.ns r.cr;.~ :.,L.ated in a genert1l fur ;;. it was t·._.1.t by LBc author thut it woulJ be of inter\::Jst tu 1.nvu:stigc1.te Lnls ir.Li..L,~nce of tLe al eron for 1'her1➔ fore ci:s 1:.:uln pur"' of this wo1.·. t.he c.:1.sc of flu Lter for trtrE:le dei:.:: r~e:J of f re~,J.o.,, ( torsion-fle;:,1re- a t l ~ron 1 is investigated fer ~ix diff~rent ;031tions of the ailoron center of ~rav.ty. com }ar,:'1d ,•,i.th. the Caso cl t 0rsion. Th0se re~~lts are then t'llo ,1~~(!,r,3..,s of frer-1doi of fle :rure- Also the subcu~e cf ~~rsion-ailfron fl~Lter is 1.nVE:f, t; ga tod.. One uf th·e coGcl.usicn.:; ;o;:.cnect is thu t the f.:..ut- ter ve ... •.. c~ t,Y ircrch::H1s a.1, .... st. ... 1rn~a.:: ... y >',itn th::, dl.,tance of tho a i _;_eron center of gravi tJ .f:':.;;·,,arct ir,.;:1, tLc i OD:. t ion of dynur:. ic baJ.ance to a posiLi.:,n of about five ; arcer,t ov• rbnlanco, bul that r~rtner iLcreadu uf uv~rua~an ce ~lJ.1 cause only a ver) s;Lall iricre&se of f.:..utter Vl:ilocity. A calct~lai:l 2, ::-cllene for t.!Jt 0.etar.::1lnatton of the ct.bic and c:;ua.o.c~.i.·tic c·o(:if1'J.cients ls ,-,re~ent-1 • togf:tber ·,~ith s. numbHr of tunles for t,110 vari 0us nur::drical c0c~ffi.cients . The author wishes to ex1 res:5 h.:.s ~ebt of gratitude to the Vultee Aircra.fL C(., r .u rat.:on, in ;,actic1~lar to •,: r. 1 • C' • ,o c ,'. a f e l..l er , flea· of the A,~• rod.Jrw .:.i cs D·,=. 'Jr t.. ,ent, f..:,r i.~1,,l:n.v, r i~ ur.e tt:e 'r-lng l arai::et e;•s of r:.1.1. VunguarJ. . It .t.:.i.::; tc- i,e the~-.i: ,Jing pu•a.,.t..l"...,.L'!, i;._v0 o u t Lei' .. i ti~ Lhe : r·...:..0~8 t. C..Lusions 1} t v .ce&.c!leti t.hc atuve in v ..H·.;;1\,r, o.i' t.he V,.,_;_tct:! 0t-.1· .iy .lt,ci te-1 f 1·v 1 th;:• :.H·!::;inr inl: t::a t. :.,ecn ,,tAdl'ieu it: i...;n-.;.21•t;;.L cc, tlt.l..; .1 ect,1.,mvi_.. t.1.n,i . o.j,d' a Ct.; ?tO C, tn r~.LJ re t,~, t.:1. \,v c0r.;, ly t.11\:: a~\ .!.l•.L1 con.lit' ... Ct- u.il'. h,nt.! . l'tieoaore v 0n : ar· ,tn , ;)i1·tct, r of die u-~n.:'..e vrato.ry , for tne intt.. ru.,:.. n:J Le.:-:; . roble.,~ . vc •'-• !:.llv:T, ;~u.::.,i,:; 1:r,.~ iil. Lav- i , tl:L; p;:~rt ic .... lar- TABLL ~1.f coNL.r ·n.5 Summary ACt'oOO ;ledgement Table of Contents Table of Figures No~at ions Discussion of th~ s;0cific r r o b1 ~1m uno\; rta ' r.m Hesu1ts and Conclu.sions References Calculation :Jchen,e for determining co~fficicnts A. B. c. D. Tables Figures Flexu~e-Torslon-Aileron Torsion-Ailerun Flexure-Aileron Flexure-Torsion /1 Fig. l. . Parameters of' Lne airfoil-aile ron cor.1b1nation. li'ig. 2 . Variation of x ,~, :r~ and. w~ parameters ,¥1th ~ • Fig. Roots X of th~: real anct i mag inary equaticms D• against~ for FleXur-e-Torsion. Fig . 4 . Roots X of tb,a r ual anu i mag inary eGuations against~ fvr Flo\.ure-'.I'orsion- Aileron . Case I. :ti. 1'~ig . b. Fig . Same for Case II. o. It Caso I II. Fig . 7. " II Case IV. Jl,i g. 6 . " ft CU:;j() Fig. !•.... II " Cas , VI. Fig.lo. v. Roots X of the real ana imaginary equations against l for Torsion-Aileron. (All six ca ses k tnc lud ed). Fig.11 . Plot, of ¥lutt :c FrGquuncy vs. C.G . Location . o( . (.flg . 1) = o( = !.i = etc. \J. i3ta r. ce ,:if tvl'S.J..\...1.:,...1.. li, S a1.'t 0f :..id-cl1cru as f ract L,n uf Sb t-cL1.,.r--..1 (f lg.:..) . Locu t Lon of ~,tiffr,--;s~; ,. u ..i..:; u1 -!'c.:::nt, 1.vtaJ. churu HC~:,ur-e 1' l'.'(Jl, tht3 1 e~-:::.. tnt; • <J ,. 0 1 s I. or ~l~tif n~~s a~is) - l l c = d.lstaneti nf tht: i-,.. i ... er;"' nin ,c axis uft of :ni.J-ct,t, r u us r racth,r, :.,f .-,,,::. l- •n,.•.cu. Lucnt ... cm of a llerc.n h1n..;. e U)(.i. S !n , tH'CC!: t U.t L~L:... ct,il'(.; ~.,t:.:i.~-iureu from lfkiu. 1-n,, e l,~e. ~-' or C' = - l CT L,t.u.il~•...: ,;f :. u1 <'-'!.t• 1· ct _,1e1v·.L.1 -·~ fi"J.C. 1-i. vf of il .. nge ... :!. n ..:. , .,ll i·~::., n:-1 '; ·1ft ei:.._,r .. .-,-l11. ::: ::: :i 3 . . . r' ;,. i J' H ,4 :: Cifi3~ of " = act d i tJ 1.. n a .i. L'..,:i.ss ~ ;;;1• t..n i t, f ::: an <.I. '"el .m tu the u ~ ~i•..:.,n to v a1·y i ..~~ c 1: •.;,f' v1' 6 1:av .1. t~ l., cat ~1,,.,n . J-( = :!t Q f 1' (; 1· V, int; , rfa t i t..!! i_ L C·l er l ... t .i l. 'l•~.l .. i.~ t of s; l:1r:i o.C rr,, ..:u -.:1 a t:/...1...lndvt 'f:1·1•ora t._, .,.:,:.> C£.vrd of l' v.!. n ,; , octn t.i ... r, o.t LL:... rC::l°er1::nce 3,~Ci:.J.1..,n . ,1ic1rr.0t.L:r ec_, a ..... :;: l.O CUtL n of :,:; : t c, .. f' rvi C ,nt,1-: ..: l:.• of 'J ( I .J. ,; • 1-:ra v 1 t:; 1.,f . ing-al le rem l) / <..f .:1 i .1,. oro n .1.'0d "' C"'U .- !'~f er r . (.;. t.., :./ r ault..J 0 1' t c ~- ~:J·rat~ur, Ul li'.Lr, . nb - c.'..leron ·1. ,.,}·:;t<;,•!. ) .,Jrut.:.0n uf ai.1,.,:rur t,(., £• ( fi , . l.) S L:il.iC' Tt't;•Hi't O!' . ' .-l ri!';-t..1 .._...._ •...'!! .. ~fl r·cf ..! rt• e,.:i to ~ • :,tutic or,vnt v !'1; , ~ !' l'E:(! f,•,.,r:en-::. Of t,n1 t ~, ,..:.n t,,,, .£.• JL~! · C .. :i. ,Jt' .:.-:.•n;, tr.. ;._•:. ect 1....f ln..11 t.i.·i .:::: 1.n ..L•.::n~;u . . _, C rerern:-J - 7 - C0( :: ::: !JJ or torsi.,.. nul ::it~t'rne~.:; uni t s_ un ..1..cln 5 tL. ..) i' torsL;n .l ~-,; c..i ffn sr tin len; th•. 1nr, 1:1ro .. i. . (,._.:..er,.,1., :· < 1.· .;:., ar una .£, 1.,r: i t 1· stiftrnL;.5 or ,. ~nt 1n :1efluc1..i0n: , rer 1.n_ t :::, ~::.r. l ent;th: natur~ circ ~i~r 1ccq~~~c1 of torsi.n&l VitH"li ... lvn ,.,1· <>.ll,r,-3.i S le:.. ar'.)till<.1 ~, one ct ..... ~ rt;e of f 1·ec..io· . .:.n \i ct..Uii • <.. ; of nhlural fr-q~~nc; vf uf ,1.i.lHr:;n h.!' t =- t i.re v = velority of ~1 . .J..ane : U.!itl f1.ut1., er , r c1lticu Ve r .£.· VtU.1..JClt.i.l,.;; ctosl.;,1,,,.1.,,;:;::.., tii:.n~1• t .. l! f.i..,sH_.,·,;;tur .. 1..,.1- .-~.l.J..~ l' •A: flu: .. 'i \''.:!..1.1.Clty 0r the al.J..erun - turstJna~ vl;'.loclt;y, V,,. L'~ 1 ,re:->JHL'.:.> r~ L~or the f ... ,::x,,.,.;;-t .. J'•J:~r:. ,, d cri i , i.n-..t ~n- f ... i.n tir v •~.J..oc 1 ty uon i.- uf l.h. al.l.c.c r. .::..:,c., ... ,, .. . r= k - pi::il.'a, ~te.r LiSc~v. t(.• .;.!Wiec.ti: t.1HJ inf..,t,.; HtCd vf tnc ~l.J.Gron on .in~ f~utLLr. re ... ucca fr.('. ,H.:11C,Y th v,cL e B½-..a.J,. n.~ ..• ocr ul in a Jlstur.cc eq,.:·u ,·e.!i....cH .... tit-:. v e J.cr,g tn eL u.L .J. t.,) .. ave ,.'f t.: . .: ·,a L ir, t r· s 1,f L.n a 3 u • L,, l1 u c-.i x 1:. rr. L.O .. ' \.. t.. ~•d r,>,_::, in thu : :J.:,1 C"1i.: rd 1.i.s-:.• . rr•<:2· '-' (' .._, l --l.... lHl 0 to X TT • - 8 - X - C O') • f u.r.cti on of f lu tt e r f i~ e . ,le l C F L _ .•t.d'e - "...orsiun 'I : f vr F'l e 1~u ra - ;,iJ._ . ., roi, ·r•, f or .1 f or Fl exw re-To rs ion-A i l e ron fot of.'s L.,n-A l l n·cn F + ~G , cct.,p l cx .i.i f t V J C tv;. . • Wh \. w 2 1 ~ (!E.. ) w,, K .,..,, ::; :s.<2 w , z ( UJ.i I K t at1...:. 1. t t:t.l ln R~f e r enc~s l . ~ , 6 , ~ . 2 <w ) K = Rl :{ :;. w,; r,2 ~ Th e :i-ur f,ose of this i,aper is to she,•, tho influence of the aileron on ~ing fl~tter ~hi~e changing the location of the aileron canter of gravity . ~ orating ind additional mass 'f.hi s ivas o.c..ne by in<·or- m' into the no~L of the aile r on fl.OS c. G. OF /l!t.ERO/V /Tl in orJer to cc.1.Ust.1 a cr1unge in tho lo ct. t lvL of tho ailt~r0ri centi:r of 6 rav 1 r.,y . This pi·act1 c ·,~ ~as &.cio! tea since 1 t corres- Tne u~slc assuri 1:ti0n ·,,as that the tota.1. ::.ass \: cf the wing :·o r ft . 0f sf.•an, inc:-i..tH.iing thu aileron, d.:.d not cl:&.r.ge ,~lien adcti t lona.i ., .-¼SS 1,us aaued tu the ailercn . - 1(, - the aileron chord forward -0f tho nlnKe a xis. ' The ileron chord being defln~u as the distance from its hinge axis to the trailing edge . ra!JrOsents the notath, n usc~d fur a :;articular If ( \ location of the cec ter 0,.· g r·av1 ty fc,r which all the wir:.g purameters ·,!J~re ..:lotercined. by tests on the actua.1 ·1 ing ·(m ' :: C) , then fo .r any oirrnr _.osit,ion 0f t.lw <:l'.mter cf ;,;ravity the aument of inertia oq~atlon w1i~ baco~e: 22 I ~= Mrf3 b-= [ 2Z .::Arp. b - 1!> / / l,_ \. ~tb )2] froc. rihich 2 _ r t> - r 2 (1) f>o - Gimilarly for tie sLutic ~o~ent r~latiup: ,, I ,. o r.>.. - .... ?: b - ~ sin c e (2) The frequency of the ~ileron ,,1.li. vary accor.,;.ing to the relation .S62.~ :: !.L W f3 • l'('o (3 ) As an auxilliury expression ~e have : 31nce .ve obtain: m' = M .X.r-,o r ', ..., - t (J .IL! !it. + 20 ( '~) - 11 - Six different stations A&re lnvestieated for which and . +40) of Lhe hi~0ran chord. ponus to t.he casa of a;1 untn1.l1.--nceu ai;.,;ron. The Vdr iation of Lhu parameters with respect to (f ) is shown in Fig~re B. For ea ch of these six cu :.;es th0 f iut t er V t. J.oci t 1 es of flexure-torsiun-al.1.eron und torsi0n-ail,-r0n r ere c.etor- mined and con:,.-ared to the fleJ..'1.i.r€'-torsion C d.3 0 c:f UH. wing., For th~ dcterruina.tioll of flu.tt t.: r velocities 'fh(.:,odorsen 1 s< wu:2e used . eqt..a.tions (A, H,C n:.10, Hef .l) of eq1..ilibrium These ec~ua tions .represent the eqv.i 1 ibritun of thB moments about!. of the entire air.f'u jl, of the moments on the aileron about .£, and of the vertical forces. In tl't " SO e.x- pressions the total aerodJnatidc forces and uumEmts we.re obtained by Theodorsen from the t ·Hwry of n onstat.ionary ooton- tial flow. The underly lng assurr,ption o f the abc v e m~:mtioned equations is that the lar~;e oscill&tit::ina .;.re not considored and only infini t cly stf:a.l..l ogc ilJ.ath~ns ao out Lh1; • os1 t.ton of eq_u.ili briur;1 ne ed be c c ns ide r~d. case of dan I ell vi' .:1ivt:rger.t .. 1o tlon, vnly the bo.t'de.r case of lntrouucing into the equations - • 1, - .5 e 1 .t\ b 0( ;;; Clo {) :::: :3 e lo h :::: i ( i\..!.b + if1) h O C, i ( k ~ + lf2 ) • V(::rtica .... f ...1r c e s: D :: L1 :1 + 1Lll ;::. Aaa. +f2a ,,( A ar, Aan Ab« :., b f; +D~ ' A. bh A c(';> /~ t. h ,1 ,l CC( ;;:; + Dn ·" () - itself in.tu the .1. su ..... uLion - of t~-.o !3iuultrurnoi..L,::; cqL.at:(.;nn, one r eal ana 0ne i .uginary. For th0 ma~cr C<lse, i . e . th.rce degrees of fceedom , the solution reswlts n cl real cubic equation and an i~aginary quadratic e(uatlon in X, ~hlle Lhe ~articular subcases of t wo degrees of freed~ : are char~cterizod by area qLadrati c ec-: 1...H;;.tion ana an i;·a;.;in:1ry lin..:ar equati_,n in { , Thr' s,,lutio:1 of :.h~ se eq~,:.-tt i (.n~; ct.n~rt·in e,; "( and ! k · f ron· which Lh:~ u.ntn._., ;, n v dloe it/ . ml f.1ut ter freq enc y c an , be obtained. . of X :-;at,isfJi.n_; tJoth tt1<; rt;al ,mi ir;ia,,!nc.tr.,- r:,,·ts of the \ ae t,._ r~inan t.. For L:1is, the ?>.ru_oh.1.cally the - r ·t~t!i- roCf.)uu.re Cl,n~a st: 1n r.lc l tlni; X a.•.., ,dnst . kl fv1.' th.it.a eo.t,;-""thns , the cf th f' determinan- f t, : ::1 Vi ty t i. e n . .t • \..' J.. • • + J. ', • ' increa:.Hi in efi· .: ct,j ve : .:i5::i . (.,;., ..,nt vf ir!,~rt.L.i C/.i c.;' l::!S an tn - It l :1 ir. tcr:dec 1 . ) Th\:j .cegton lz ,c ,t1-:n1 . I \_ w~ \ I , .J-. ~. Theonor3en, Tl:1s..cch.,,·c: Gen,:ra.J- J'h or;;• of fH::l'o(.;yn;,:r.ic ln:.-;iaoi.LitJ anu t..L: ucraniJ:r, of t'.1.t..t."cr . I . ,·1 . }l v • tl }_, t. , : J • 1\ • C . . ~ • , l ~.; .. , ~- 1 • Theud.orsen, Thi.:udo ce unu 81:irric.i\ , ~. E.: i-:~cl:ani s 1;1 of Flu t, t0r . '1'1k. ,re Li cal [;.n 1 ::/4.1- t:ri r, er. tal I nvest 1.tJut.ivr, Or'. Uit: F'~utt0r '"l'~1L1 .H:l: . '1'.H . No . 6c1b , H . 1\ . C . A., l.,,'-lv • .." . Lor:,b~d'u , A. :~ ., Sr . : fln inv e~t.i.!',ution of tho c cndi tions for UH; ucc1.,u·Emce of f :1.,. Lter ir; aircraft ar.d the J~v ~lo,mont of criteria fur thG ,red i ction and e.i.i . ina t:!.c.-n or S'..i.Ctl f ... ut un . Loctc.r I z Tl~es is , CaJ.. lfo.cni ·l In ) Litt. te c f L·cdrnoloiJJ , H·3~ . ~• .<.as.::ineJ.' , ·1. ., and Finr;ado , H. : Dc1s l!l,ene ·,roblcro de r Flu.gel~wm.-ini, 1..:nt; . Luftfahrtfo r c.hun l:, Bd . l Z) , Ur . 11 , ~-- :~ov. 1 ~.s6 , :-; . Z7q_- ,..J:7 . F'r.~:ler , ~i . A., ur:d Lune an , ,; . J .: Th::: f'lutt.0: of .f..*'J r o - : lane IinJs . 6. 1 • an- • N~1 . 11:.:,b , 1\ug:js t l0;..;c 1• 'Fr~~ze ·, , . A •• anG Dun c B-n , L . J .: Tht.i Flutt>2 r of i'cno l1mt1s , "Hrianc::; ,L <..t Tail l,ni Lt> . (i\ seq!H1l to ;: . and' • No • ..i..LL.,,) H. an .. • • 1:..,;:.f:i , ,.hnuary l,~il . ' - wal fc ua ti en Th 0 fo.1. .... <.1dnr, c.J..e ·en ,,,in~, .•dl'b. Pt_rs . ill. h ::;. ve t.o L), a, ; ,r., c 0 ,:..;re ;3c ie: .e . C1 ,.L.l LH.. fc .L . o ·; ,eJ throut.J 10ut t,h.i !3 CoJ.CUlation a) A~. =- _ti,:'k IT K. A/!>, = ir [ - :c-;. -(½ - a ) T4J r :: Al~.3 = Ah. 1 - o1, = B"' 2 Ill ' l ~ - a r z .,. - K. ~ I< l rr !,7 TT - ,~, ' - rr•1 ,., l l ;;:: l T~ Tu - 2Ff2 J.. n• ( '1'5 - i} h1 ::: x~ c.,.1 ;; ii- T4 "' l 10 )' K .!. 1 ~ - a (- Ah,) K l C ~1 = ~ C132 = - cf!,3 = V Ch, : l K + 1 i'I ½l 4 ) - - frl. ,· ::: cd.z 1T i j = B 133 ..,, ft\ - ~ K - ii'l K .J.. Tf T4 I', ½) - .:.1) (= ."-,p) - a) (~,K + (c ( l (;:.Ji.->.,<- - + '1 ,0) (1~ 1T = __, ~l B f?2 - .iL. + (c-n ) J.. TT '< p K '1'.1. (::: I.hi) T obtained frc.J,, ,1t·0 the on .i·ag e 16 Ref. l. I a"' ;:;; 'l • b~* u. J, L - l 1:1 = 1 ~ h. ,. :1 (>, ./~ <12 - + t'Z. b~• ':i \ ~G -(½ + <.;) ;.,- -(¾ - J) t;J;i' ] " r'., ~l +[~ r;' , , ~3 ;1 .,, - ..., 1T ;., TT ~ ~ ." !:::..:.! + lS. ,, H Ch = - C ni + H eh Io, = J, l<. ::. ,:F Flex~r o-lor~i0g- Ail8 ron. bl = fl 2 :: tJ 2 :a I' a ,> ~ b r, JO d.1 = ·' \' •• P.1 - t'._ .l ~ u) -C½ a ½- C cx1 - ( (_l; • ' h1 t: :.) h1 C h1 . ,:'"\ ~~ Al,. ' '" J:.}! + ..wl ., TT 1T ·JJ S a ... ~ s b -= A h,. =- .1.Cl'1, 'h1 "z (' h, C ""2 ~ _, ",-,, C ::. Const~mt /'\ d.2 ;. Cod'ficii.mt:s : (¼+a ) { C~Jr;,) - u ~1 (c,) ~ ~ (~)· } - Tu. J A'(d - c·•,r'i + J;.. :i.Y.Tf tt) ~ jT l o ~1 !, ( ,. + r.t / { C />1 1 1 ( r,;\ .,) • .tJ ~l ( ScJ ' - ..!:Jlrr { C~J 1) i r;3 ( !, ' C' • } } I J , f; ~ 1-2 l " ,. \' • 1 • +. ",-2 \t u.f ,,. 1 •. t~ .., 1 ( .·. •2 - :~ h :: 1[ - <L + + t ~- ...... A + ; 2 : ; -AO\ L ..L k 2 Cce ffi c i ~n ls of X 0 ./-i l L) l c 1 D1 - 0 1 :;: r ::: A 2 - \., (li ::: b ~• c.,, = - ..:· /3~ C"• ::: L 13• - ~ s.\ = ) ;J. , c. (' •r .[) 112 'l'u ( . ) t lT '· IT Cfl• + -•J l2 ~~ B ~~ I - : rr . \ - 'LTia\. L, • J,~• C2 == ·' 12 ( ' ~z ,. '..,' ')'2 f.... L2 .~ ~i !; 2 ,· a1 + + .:.l a· = - ~,~ A2 ,. l + \ ~ + b.; I ~ ~ ; ' . ~2 ' lit - ( ~2 / ,., ;, h1 I \ ' rz J , , -7•t , ·-1 ) j-~ 1,· ( Uj n. >.J ·r'+ ::. ~.,]? ---z .r:<. .- ;. 1 ~ ...) . .I 0..1 l ) I ~ ~ ,~ 2. I\. ,t U/ (C J * + ;- -1 + + ~ (c ; ( ~(} k 2 ( \C I '.- ] (I) i\ ;; - (½ • q, ' .a.li-/ .. , lT \ ,, I n) ( C) c* :::: - (t 3 - = (' ~ / (" \ 1 / . ~ I, Iv<. JI ' ·k = La Ch, ; . rr l' " h1 \ L.)·* -- - \:~ + aJiJ {Jl ( v f' :: '" ~ .., iT ( q) = - (½ + a) 11~3 - tT,2if A ~3 ( ~.) :::: 1',i (; ) :: (t j , l. :J iT _o.Q( A2 + 12,, A;· f • !i- Aa. + (½ + A ci2 - .:._;1 TI l~ .... A ·!( J-,. ~l ) ;J ''~1 +(½ + o.) B 012 - (¼ + u)H~2 - "'-'iviT Ar.32 = 12« :i. ;· 1-" Jl~ Ai + J2,, J", ~ :::: ' 1*2 ;:. f2o: B ~ + l1~ •3i ,j 2 "" J1o( I 3 = 12 J 3 h 2 + fl,, + 3 3 .•:. Q( c; +fl.r-, C' i. +flh c~ = .n. . . c\ .. +nh D~ L I, Jr I f2 h 11 ~ ;;: ..n."4 D2 +fl~ u l. - = 11h L 3 J1~ ::: i '.¾t / ( ~ ;2 ra Wo, I W~} .J.. I._ ) We. ? 0: ns a 1·e fi na .!.1:,, g i v en by : 0(.u.a t i r.- A Cl -= @; :; I 1.,. I 1: ' .ri. IL + J, :bl+ "-; \n) [ I , " + J, I '( + ._.; (L,) .I\ 1: I - 2 1- + ~\OJ. : i{ - .. l,,.)' ] !' uu.t i 0n ' l.... ,.. .,. \ \ 1.. ... } :. - [ J1"' fla ':; 1 + 1113 (D« (\ 1 + J l h A~,)] =-J2,J2~(-·G) + .fl r-,flh !1~ + Jl,,Sla_ ·, ~'r, L. Cc,1e,ff ici ,s nt of ,w uJ. . t uuticm j.,. (IV) 0 : 0 {1 Oi Co~:ff 1 C Ln t GI = 1 ,, ,ff Con:., Lunt +- ~(b) i, ','ih.8re : (' c ~ff i c- 1 .. n t of < C',, eff i c i _nt \ 2 (.;l fi 1 = f2.r.,J2h ,- .. Cr'(:: n~: l. flr,= r'on:;, t.:.,n L ~ 2 ~ = 1L. [ - 2t i Ctt >2 r; - Coeffici..mt. of : 7 - 2 Coefficient ol' X. Constant: bagina.ry -:g uati n: Coeffici0nt of X Con.tant: - ' ., "' ] .n Case I, wh ic h c o,~ L' Ern r, onci s to 'f = G - 10~~ Case II, " " I! £<[ :: - Case III, II II 11 ~ = .j.. 1. 4. ,,:,, G z = + l \..I ..~. T IV, tt II rt Case v, fl " n Cuse VI, !I !! <I Ccts e 5; + ~u% T = + 4,~; ,k- .£' : 1" (J The fvlJ.O•· in~ ... i ns :·nr;..: .. c.. ters , cor res ;- on..tlng to ;. 2 re Case III, L.. ..Hl u. : a = - o . ,.A 1-( C =0. :,i -::. J . b6B '[ = o . ;Ju O l,; :. l IB:5 mo= 0 .111~, b :, 2 . 6b , )tel. ::: 'I XI> o. u~ = u . uou'/176 = l CO r ad/ .;;ec. w w.,. z: !.-;<lv r :ld/Buc. ,:i. w~ = l u;:; rud/s vc. The vari a tl DH of chc tr:r 1Jo ai ;. t::ron f !.ir1,rnrnter:J L; IV :!DU V :r I g:'r \ -J... ~..,' 111 . I"J V V! + .; -· +~ .,, +.i. u . v.: l...A h; . v~~~~ 'l B~ •· ?b .a . ou u7 l't . O...M&~<• . 0 \..,1 5~ . . i 6 • Gv8 ')- ..l .~· ·,1;:.:S..:. • .:vl 2 3 i1 • l .:20 • t --._,\~ 7~1:4 I (ff' .v ~ 'I.J.. 1 ,;::, x~ • UutL:-' vv ( r 2 \v I • Jl :.3b .:: 1t . \.,l l ::.. G.... tJ!_ • ..) ~\, ~v . ,. ~ Wr., • \,L.J... i... ,(.... -- l • ~ + l. • •; ff, I lt,1 0 . :/l 0 .. ! f' ..l v ~ 117 . b l -. J •;; l;.;11w • .,,_, 1,10 . 8 f......·t.il, • ) ~ . ,, ,.;,~ • .it:!_ 40~2 , •. . ., . . ,..,,-' ......r.., .,,,,,,,v • . 4 ,.:, 744 • 4.C 'f ,~4 • •~O'l•-t4 ,. •,h/74.4 • ➔<·, ·J 744 ~ ,) lo., . · J ~-~ . 3~ ,2 . ' -·.,, 1 J?4.., . . 1 2'7-.. --.; . vr .. . . . . . .). •. 'l ~-'...... ·~· ,) .;.,. V r • t..~,'S .. ?(. . 0~4...:868 • () ..~1·161~ . c~, . . . . . =...,v • 1 .'r' 111 .l ._ \,.., •.• • ,_ .~J i . .. { ~· , ..,l~ • .,.......,,.,. 4.J'--., · '· ;,1.1..n • .52 ' ,;~).C - ~~v1.... ~ \,,,,' ~ • ;...,~~8~~·~. C) • ...1 .... ..... lr.; ., ;..., • 1 l l 1 ......, l .l. l • l eC: 19:-:, . 14;~;';t3 . 1 ·1...,,;I; C ~a 0 v C li1 .i.t • b .... rl • :,; c ~2 . 1,! ~v,, • l -,:!,58 (; V ..1. ( • ,.,, l r . Li '"l , . l i.r....t):..:, . 1 --~~ g C 0 1'1. :· l ~i" . !J ,•:1. -t,#•"'-• • 1' ,J L t. I I l. II I IV III . H·_t;ti 'I • ~,'z • 0'.. l 'lcG . 11.1'1,1 • v 1 . 04l.1..- ,../16 • tJ •Hj •..il .L • • l.Jt .... ~ ,J~ .._)..__< • _. "'-1U~~ {j • 0 ' ;_., 1. ,, . ....... .;'. • (i<1'2 0 •,, C l . ti J,.: : ~; ~' 1 . 6 ...~ ,- ✓ .J l;) . ... 6,Sb ,1? R l • .:.H5 ,.1..: .J . \; j. ,.; ;.;,i->;4 • !../t r/ ;,..:;0~; s* • G,A4 7 '.J • J,, <i. v i •. q -;~ '- · - ~} (I ,; * • .tr{-... cl{; !.; • v ' '1 2.b VI V ... 1\..,.• 1_ .I. ·4 <.) T* . 0,17 0~.1 • u . u 1,, ...;l • .J..0J...'1~:~ • ..Lt> i.l~•7 . 1 : ,.. C-~.; t7 if ¥- . l'lcl l v7 . 1,~u. ....... --~ . .Ll:2!3~~ . G~.J..7 v ? • 1 u ➔ !- u ~~ 1 . d G!;:Jt-'lt.,() 1 . <j,'7 ~·;u,:,u J. . lbob6 .j • l,·ic.! 36.:.i • ~- ' 1 ..... \. i,b.., u • l'.0', ld 2 'i"/.·4 1 1 i~~t: :: l' , ., • c: ,._ fi r. ( I '-' ,, , (3·, ~:..J~... t' l ~lf; • ,.J6 ..l.. ".. ).,,_ ._,,JG • 7.:., 7 ,' ,c • • Ll 40 ~ J,..{l . .;;. . lul 7 . ·J l6llG • j • ~} l • I', ,.1 ""P , .... l I II lII V A l H • Bllt..b 1 C1 D 1 .-, A 2 A2 b2 TT . Ol6bl •' 2 C2 •.::, . B4', l D2 >. ;., 3 r( "\ 3 ~ r3 lJ . ,, B3 I.I· C3 ,... • ·--• 3 r. ~3 !].:, * • VT Cases I II IV III V VI C.v2 * • 0 7 d ?ou7 • U00::> ..~6v • uG:._:t1l 7 b • OV706.;B • ·:.., ;,,4 l--.n1 • C,L\ , ,!;87 I"' * • ;:165bb.~ • "'/~548•* .6b~k,j ... • 5007 ... ti . 4•H:i~l6, • ..)-~ 1.;tj ~ ,'.I J"'I * • Jcl0!.:i~ • iA.;:7 :... 1 • .;U:37~.iJ . .:, \~ l ~~ . 2'- ?J,.\1 .. • ~fAl-, 1 .,, .,. l.z • (J 74 -,,<lb • OL/; ... !.)·J • U,, 77bti • G~dJ7C . O lB U➔ l . 0 ,. i:J'•i 71 J2 ,.. ,.. • ,,; l .6 7•=~~ • 0 L~~ ? 4 ~: ;5 • ' l,:ls.:.i : . 0 l~~ 74~: • ()l ..~ '7 4 :}2., . 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T - • I1 ''f ;: 1 ,t -1•-;-· -~ -- . -'--· ,·t: • : ' 1·T' J!, rJ-t ~~ _,-,, ;~ --· :~2:lii1~~fi~~~~•kW ~ 1~f, i,)~~iit0~~ ; ~:J§:~ 1 J..Lr :-1'-_L ...:.-,'-:. c....j.- - + --l . .+ t ·- i--,-- ,._,· t--+---+--+--- + --+- -+--+---,t---+-- -+-- - t -~ + --+--t-~"l-.-;-~-;-+-- _-;-+---.,- ;~_,~ ._ t-,,-.•- - ~+-- --+--t-J- _- _+ -~.-_;:_+ -.-.---. +-.-LC-+,- t- ~; t .~ t' --r ·I '.;.li ' ' 1' - + -- I • -;- 1 I! ,'.'' --+--l----+-'--+~-+--+·..+-~-<-~+-~-1---·- -t i ~j_ -"'-'-i . +tC:" ,~ 1 ~ r •-r ,__ t, ' ,:r .. ... ! 1 J ; ' • •"· I-;-• ' ;,r • +r ; r ;'J:~ {' ,( • ' ,.. ~.. ,. . I- ·1 • • ~k-J-!.~-~e----+- 11"' M" ,- , ' "" :, ~ 3 ~ < .., 8 "'. II, : C ~ ~ gJ