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I /I '6 NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS TECHNICAL MEMORANDUM YO. 971 THE ELLIPTIC WI'G BASED ON THE POTENTIAL THEORY* By Klaus Krienes SUMMARY The present report deals with the elliptical wineg in straight and angular flow on the basis of the potential theory. Conformably. to the theory of first approximation upon which the calculation rests, the k.:nown requirements regarding the shape of the surface and', its angle of attack must be met. A further condition is that the slope of the surface toward the streamlines must be a continuously dif ferentiable function of tie points of the surface. If this is not the case, in a given example (for instance, by aileron deflection or wing dihedral the latter being of importance in sideslips), the discontinuities must be replaced by suitable rounding off. In general, the cal culation of a given elliptic surface requires a series of .infinitely many potential functions, the coefficients of which are afforded from liLear infinite systems of equa tions. The expansion is stopped with a certain term, de pending upon the degree of accuracy desired. Its effect on tie integral quantities, lift and lift moment, is prac tically negligible. .n immediate prediction of the in duced dra,. is ruled out, since it would involve all the coefficients of the infinite number of potential functions. Otherwise, the lift distribution at the wing tips does not approach zero or the do'.awash becomes infinite, which is due to the fact tiiat the load distribution of the lifting line is developed here by spherical functions (equation (80)) which do not approach zero at the wing tips as do the trigonometric functions employed elsewhere. Dn the wing in sideslip, which can be suimarily replaced by a lifting line, the socalled parasite drag (reference 2) 'Iw F = J /(Pu Pob) 7 d d y *"Die elliptische Tragflgche auf potentialtheoretischer Grundlage." L.f.aM.M., vol. 20, no. 2, April 1940, pp. 6588. 2 NACA Technical Memorandum No. 971 would have to be defined, first and the suction force on the leading edge subtracted therefrom, where, however, extrapolations are recommended because of the finite num ber of computed coefficients. Even the,resulting pres sure distribution is only conditionally valid by few ex pansion terms, especially near the wing tips. It may be mentioned that the computed potential and downwash functions change on transition to K2 > 0 into Kinner's functions for the circular wing. A large portion of the computations were made on the calculating machine, the accuracy of the slide rule being insufficient in the calculation of the elliptic integral for higher n. INTRODUCTION This article is intended as a contribution to the theory of the lifting surface. The aerodynamics of the elliptic wing in straight and oblique flow are explored on the basis of the potential theory. The foundation of the calculation is the linearized theory of the acceler ation potential (references 1 and 2) in which all small quantities of higher order are disregarded. This affords the following. simplifications: 1. The z coordinate of every wing point is neglected, i.e., the variation of the potential function corresponding tothe pressure pump on the sur face is situated. on the base ellipsoid (fig. 1). 2. The streamlines, along which .theconvective inte gration of the acceleration is effected, are straight lines parallel to the direction of the .stream. In.the case of the elliptic boundary, solutions of Laplace's differential equation A I= 0, as products of Lame's function, are known. The acceleration potential and Lame's functions. Suppose that the elliptic wing is in a stationary parallel flow with the velocity V. The fluid is homogeneous, in compressible, frictionless,ahd not subjected to gravity and nonvortical outside the lifting surface and the shed ding vortex band. Then: .iJACA Technical Memorandum No. 971 3 = rad p (1) dt p The pressure p is expressed by the potential func tion \f p p, = p V2 (2) where p,._ is tne static pressure at infinity. Function 'j satisfies Laplace's differential equation L = + + = 0 ( ) 2 2 + 2 ex :y3 oz and may be visualized as being the result of a superpo sition of sources and sinks of intensity 'YXFp, LF, zp) or every point of the surface. u(xyz) = ,(xF,yF,z ) L 15 F (4) o: \R, where n denotes the direction of the normals of the sur face in ( ., yF,2 ) and R = (. x.) + (y y. ) + (, z \) the distance of the starting, point (x,y,z) from (x, yFsZ). The integral can be exchaned for one taken ever the surface of ithe ground ellipse. \j(T:,y,z) = ,' 3(X.r,y). d xx d 7, (4a) The mathemriaticl:. treatment of the present case be comes possible by the introduction of ellipsoidalhyper boloidal coordinates (reference 3). The semiaxes of the base ellipsoid being c in direction y and c il K2 in the direction of x (so that 2Kc becomes the distance of the aerodynamic centers on axis y) the new system of coordinates is: X2 + y2 + c 2 C p a p p 1 xa yE za + = c > > _L, > K >v >K (5) 2 2 2 2  2 2 z2 2 + C K 2 2  K p 2 1 p 2TACA Technical Aemorandum No. 971 The solution of the equations leads to: C 2_ )2 x 2/= 1p2 K _K2 46 K2 K z =vp 1 21 (JI77 I K2 (I , y = cpK (6) The surfaces p, 1, and v = const., respectively, are confocal surfaces of the second degree; p = 1 yields the elliptic base surface, t = 1 the plane z = 0 out side of the elliptic disk. The surface element of the base elliosoid is dx dy = c 2 2 S v _ (7) besides which c ^19 1^ = /c y2  S K21 2 K is valid for p = 1. The introduction of new coordinates u, v, w (8) Y(u) = p2 1(1 + K2), 3 Y(v) = 1(1 + K2), Y(w) = 1(1 + K) 3 for p, m, by means of eierstrass' Y function erence 4), appearing in equation (10) d'(u) = 2 /(Y(u) ei)(Y(u) e2)(7(u) e3) du (9) (ref (10) where the quantities el = 1(2 K2); eg = (2 K 1); e3 = (1 + K2); el + e2 + es = 0 (11) i.e. , YNCA Technical Memoracum 7To. 971 ..'e e3 = '/ K 2; /e2 e5 = ; ~/ e = 1 ere posted., gives S. u) d u =   S2J/p e,/p e2,.p e3 and Lalplsce's equation reads r. /dp d p (12) :I %/L47P  [Y(v) V(,I)]. + [v(v) Y(u)] IL+[Y(u) Fcsting the solution in the form 'j(u,v,w) = 2(u)E(v)Z(w) gives, for each of the three functions 3, ential equation Y(v) ) 'V= 0  Tv w 0 (13) (14) Laplace's differ d2(u) = [ + Yu)] E(u) 2 A+B' uu (15) with the separation constar.ts A and B. After posting ' = n(n + 1) and v =  1 + B 3 and again introducing o, u, o by means of equation (9), Lame's differential equation (reference 3, vol. I, p. 359) reads (p (.) ( i(" ) + 2 .2 1))an " + ~ +K cv u(n+)> (.., = 0 9 (15a) For B = n(n + 1) there are precisely two 2n + 1 values Vm, for which En () has the form (reference .3, p. 360) S.V % lv ib5 (a0 1'C + a2 1 2 3+ . ,'o 10 CL . (16) NACA Technical .Memorandum No. 971 Onm(L) = ao0 1lE2E3 + ... is an even polynomial in p of degree n C 2 EG. The values vm are dissimi lar solutions of an algebraic equation, resulting from the condition that On (~) is a polynomial. With a view to calculations later on, Enm(v) is defined as follows: Enm() =1 41 U I i ) 0 3 m L K LK = AP (cos 9)2 (sin )On (m) (16a) by putting  = sin Cp; = cos ( ; DA = 1 sin P (17) K K The solution 1 nnm (0,p) = nm ()Enm(V)Enm(p) (18) achieved with equation (13) can also be represented in different form (reference 5). For, on denoting the zero places of the polynomial Onm(j) with Ps it is readily seen from equations (5) and (6) that Enm () nm (V )nm(p) ScoXst y ( t2 y2 2 a c t= ilyyz xyz I  + c (19) c z zzx psakps 2 K2 Ps Ps2 I One of the factors contained in the parentheses is selected and the product II is formed over all zero ps. places beonging to Onm( ). The zero places ps5 can also be determined direct by applying Laplace's operator a + + to the right side of equation (19) and making the result equal zero. The system of equations for ps is then as follows: 3 2 33 31 4 s = 1,2,... + + + 4 = 0 (20) Ps 2Ps Ps 1 4 s ps Pq m= n E 2 s3 ZACA Technical Memorandum No. 971 The potential functions in the respective form of (1i) and (19) are socalled "inner" solutions of the po tential equation, not suitable for our purposes, since we require potential functions which ordinarily disappear at infinity. These are secured by taking the Lame function of the second type in the variable p. This is the solu tion of Lamn's equation (15a), which, for p > co as cost >0. It has the form p.+ P e integral can always be reduced to elliptic in tegrals of the first and second categories only. The aspect of the outer solution of.the potential equation is tnen as follows: " m m (E ()Enm(P) r do 22) *Jp [Enm (p)32 P21 V/pp K and equation (19) yields ,m const1 x _xy ( x 2 y 2 "n (xy,z) =  ly yZ xyz X + c z Zx Ps 3 ps + z c2' d (23) 2 / r 2 n / P K 2 ps From the representation of :he potential function by (4a) as sourcesink superposition on the elliptic disk, it is ?ppsrenit that the potential functions in the plane z = 0 in the outside zone of the disk must be zero. Hence 1 containing factor z must be taken according to (23) i.e., only such Lame functions as are of the form (cf. (6) and (16)): E m( ) = P2 Mnm(o) (24) Lame's functions have orthogonality characteristics similar to those of the spherical functions (reference 3, vol. 1, pp. 369 and 379): NACA Technical Memorandum No. 971 \ 'd 1 : .... ^ .O, "if m # t S n() n (25) J ./ n if m = t 1 + m t  P E nm(P.)Enm(iV)Es ()E t()V dpL du J J 0, if n s or m t = or, t (26) I, if n = s and m = t Tl connectionn with equation (4a) is established by having recoulse o the following representation of 1 (R =J(x XF)2 + (y y) + (z zF)2) (reference 3, v. .I, p. 172) 1 =. n+1m(,)E () Fm(p)Enm(PF) nmF)nlm() o > R 2" c nFO m =i (27) whence B R/ z =0 ) T, I = 2cm nl nm Enm mP )nm(PF)Enm( l  2c 1p.y 1p leaving the summation over Enm(p) in the form (24) to be effected. Then by assuming that the sourcesink distri "bution on the elliptic disk approaches zero on the edge with the root from the eice distance, i.e., with 1iF , '(xF,yF) can be developed conformably to products of Lame's "function of the type (24): CO (T(x,yF).= E Ast Est .( F)Est(U) (28) s=1. t  after which the foormulation of the integral (4a) gives, based on the orthogonality of Lame's products (cf. (26)), IACA Technical Memorandum lo. 971 9 Srp)= A /MM Annmm(1)E m(,)E nm(.V)Fn (P) (29) 2 n=1 m .cuation (29) proves the potential function (4a) to be a certain linear combination of the functions Vn which are analyzed next. epDresentation of t.he potential function fnm as a definite inteEral. If X = 1 Y(t )e v'y()en .'" (v)e, vy 'V (w)e. ea.e ) (ea..e.y (30) (eC defined by equation (11)) or if (6) and (3) are taken into account / ','(t) eg x /, .t) ex y e (t) e z ,/e2ev veae., c. /3 ee. ve3 c Zee e1 3 c it can be shown that M( ,) E; I (X) m() dt (d) 2n i +i n 2 i T1 i, e e e z  x :,o: nial [Y(t)] 1 (33) if loop about tha points of the complex t plane cor respondin. to X = 1 is taken as integration pat.h from Tn toward r 7. Qn(X) satisfies Leeenire's differential eaua t i on (1 X2) da x(x) 2X 'n) + n(n+ 1) n(X) = 0 (34) dX2 dX and Enm(t) complies with Lamre's differential equation (15). Then [ACA TechniCal Memoraadum No. 971 ^n(Wx.) 'Rn (x) n(n + 1)[Y(u) Y(t)] Qn(X) (35) 8 us 3tn for any two variables each of t, u, v, w; i.e., Enm(t) i x) Am + n(n+ l)Y(u)) n(x) L 8u2 Sm(t) a2 Qn(x) d,(x) Enm(t) (36) a at2 0)tm ( and a2 2 (nuvw)[A[ + n(n+ 1)(u)] jm(u,v,w) 1 r m 8 n(X) d Enm(t) ](2 iE (t) (37) 2Tri Ln a t a t JT The same holds true if u is replaced by v or w. The integration path is next so chosen that EL nm(t 1 ) E tm(t) 1 = 0 at dat Ti i.e., '0nm in every one of the three variables u, v, w, satisfies Lame's equation (15) and is accordingly a third representation of the potential function defined by (22). The points X = 1 in plane t are given by t = v + w + u (X = 1); t = v + w u (X = +1) (38) On the elliptic disk (p = 1), Y(u) = el; i.e., u = =W (39) The potential function 41 is cited as an example. The'sole Lame function of the first type and first degree equipped with the factor /p2 1 is: El1() = p l 1 NACA Technical Memorandum No. 971 According to (22) and (23), respectively, the poten tial function then reads, respectively: Sdp : / (p 2 1) /.I 1 o 2  p and jP Lift and lift mom:ents. The lift is given by F = ell Pu Pob)dXdy = pV ff (V'ob 2u)dxdy (0o) whereby V nm n)Enn) n (1) = =E m(u), n ( ) Sp= 1 ,: nm 1) /( S 1(41) c 1 (1) 2 41 Base on the orthogonality of Lame's functions, only S1 contributes to the total lift A R V2 F (42) 3 2 ell 13 furnishes the pitching moment about the y axis 'A = i c R/ FelI (43) ' t.ie rolling moment about the x axis L c2 V ell (44) (The negative index refers to the odd functions in y.) The elliptic wing in straight flow. Assume the el liptic wing in a stream in direction of the positive x axis with velocity V, l'ow equation (1) enables t'he cal 12 TACA*Technical Memorandum No. 971 culation of the velocities induced by the pressure poten tial \ in space and especially on the lifting surface. The z component of equation (1) reads in the stationary case Ow aw 8w 1 p 3 8a (4\ (7 + u)  + v + w =  (45) ax y p 8z Pz Small quantities of higher order are disregarded, i.e., = V  (46) ox 8z The z component w of the velocity vector w is hereafter called "downwash" for short. The downwash on the elliptic surface is obtained by integration of equa tion of equation (46) for z = 0 and y < c over x: x w = / d x (47) V / oz The calculation of the integral is readily secured by hav ing recourse to the representation (32) of the potential function nm. After formulating n by differentia tion below the integral sign, equation (47) gives x w m 1 / / (t) e x V 2Tri I / I  ., J e'eg e1 .es e3 .+ Ve (t) e3 y v+ (t) el z m( dtd /e ei /e. eg2 W Ie e2 e e3 J (48)9 Now a3n(X) _d(n(Z) aX dn(X) 1'Y(t) e1 1 az dX z dX e e e~ c _n(X) d %n(X) X dQn(X) h/7(t) e2n 8x dX 8x dX /Ze ee Ies e3 c NIACA Technical i4ewdrandum 1To. 971 that is " _n(X) 2 (x) ,,Y(t) el (4 0 z a x ./Y(t) e2 which, when inserted in equation .(48) and followed by in tegration over x, affords since lim Qn(X) = 0 for X   wm i r e/"(t e 2 iK n(X).t)dt (50) V ( t ) e2 4, (X) is given by Ieumann's representation +i nY(x) = W, ,i (51) 2/ Z Y 1 Similarly X(t) is expressed with (S) 'Y(s) X Y) a, y Y(s) e5 z Y.s) = ___ c i~ .+ (52) .'e2el,,'e2e '. e3el%,'e e. 6'e1e2A.'e e. whence (50) becomes w I 1 , P ((s)) dY /':(t)e1 (5n) Wn n is  t)dsedt (5,) V 2ni 2 .(t) (s) ds /Y(t)e s The integrand has poles at t = s, because X(t)  Y(s) = 0 and at t = + i'.: where (t) e2 has a simple zero place. The behavior of the denominator near the zero place is defined by Taylor expansion. It affords ./p(t) e2 = [t ('r. +iw3)] ( ./p(t)e) .l+iw + ... = [t (a + i iu3)] j /e e1 'e2 e + ... (54) taking into account equation (l), as well as (t) Y(s) = (t s)' > +... = (t s) + ... (55) C st/t=s os ITP.CA Technical Mermorandum N.o. 971 according to equation (52). Then the integration of s = v + w + v. as far a.s s :"v + w u corresponding to Y = 1 to Y = +1 Cequation (38)) gives 1 P Pn(Y(s)) dY = Q(I(w + i W'2)) (56) 2 / X(W1 + i 23) Y(s) ads By / P (Y(s)) l(s e nm(s) d s (57) it is to be noted that the integrand for p = 1, i.e., u = Wo has the period 2,1i, so that the integration path can be shifted until s proceeds from i W2 f! to i c2 + wo (fig. 2), which corresponds to C =  to + , when IY(s) e, = sin E that is, ds = d_ (AE = 21 K sin3 ) Moreover, let Ac x = = n (58) C'l Ea C so that, because of r Ve2 e X(wi + iuj) = ((i+ i+a3)= e2) (59) and, according to (33): Enm(u i+ U )= i BEnm()_ (60) the downwash function on the elliptic disk becomes w m E nm )()i Pn Qcos + nsine)Mnm()AE d (61) V n2'bI CHOSE TT 2". .. The coefficien'tsobtained in the _'olynomial of ( and n are complete elliptic integrals of the first and second categories. The calculation of the downwash function in the case of n = 1 is expressed as: IHACA Technical Memorandum No. 971 E .A) = J/1 As , Pi(Y) = Y, ( From (61) follows w11( = 1 K2 Q' (n) 1 V 2 that is, Wi' (r ,'7 0(  V M11(C) = 1 +77 x) = In + 1 2 / (( cosc + n sin )  d C C cos C 2 T:e lifting surface. The shape of the given by z = z(x,y) sraced i 0 surface is The slope of the surface in x direction must agree with the direction of the flow at the same point, that is, az(x,y) = w(x,y) cx V (62) from which follows, for z = z(x,y) z(x,y) = w(xy)d (63) the lower integration limit being arbitrary; we equate it to zero and add to the value of the integral an arbitrary function in y: z(x,y) = w(x,y)dx + g(y) l (63a) ) .I 1n L % The lifting line, the induced drag, and the suction force. *ie merely refer to the corresponding chapters of TNACA Technical :Memorandum No. '971 Kinner's report (reference 2), where these problems are treated in detail. The results are readily applicable to the elliptic disk. The lifting line, by which the lifting surface is assumed replaced, is obtained by cor relating the lift elements through integration parallel to the x axis: +xR +xR anm() = (pu Pob) dx = 2Sp nm=1 dx xR xR It affords, for instance, o 4 c (ri ) agl(n) = 4 2 a,(1n) 41T P_ V 2 K ) (K2 p) (64 a4(Tl) .,0 The potential function of the second type. The fore going potential functions lend themselves in any way to linear combination and yield the corresponding linear com binations for lift, lift moments, and downwash w. The po tential functions dealt with so far afford the aerodynamic quantities of a correspondingly curved wing by shockfree entry of flow, that is, at a certain angle of attack where no flow around the leading edge occurs. The'arbitrary angle of attack is obtained by superposing a flat elliptic disk with its flow, where, as is known, the leading edge is suction edge, that is, the lift density approaches in finity. All the potential functions of the first type approach zero, however, on the disk edge with the root from the edge distance; hence the task of finding poten tial functions that have these qualities. They are achieved by applying on the potential functions of the first type at constant x, y, z, and K2 the following boundary transition (reference 2): S. d= d 1 m(xy 0,sc)] (65) n cn i dc n :IACA. Technical Memorandum No. 971 These potential functions on possess the quality of becoming infinite on the whole border of the disk. Because of the condition of smooth efflux on the trailing edge, iz later is necessary to combine the functions linearly. The downwash function of the second type. In con formity with equation (65), the dow'nwasn functions are obtained by applying the operator 1 d cn to equation cn 1 dc (61), while observing the interrelationship: 1 d [n d Pn ) d Pni(r) cn 1 [ Pn(n)]= n Pn() i Sd d d at c The same applies to Pn(Y) an&. (n). Equation (61) then gives +a 2 wn mi (r:d qni 1 d Pn(Y) M M( d V n d 2 ./ *d Y n cos E a (66) In dealing w.ith the second fundamental problem, that is, in the calculation of a prescribed ving, potential functions are used, the downwash functions of which are independent of x on the disk. According to (46), this implies, since = 0, that E" 0 (67) Sp=1 With coefficient bn still to be defined, we put _n(x,y,z) = L b .nm( ,y,z) 0n(X,yz)= bnmn n (x,y,z) m m (68) Correspondingly it is: Wn = Zb Wn mi w n br wn ml (68a) Wn(x,y) is designated as downwash function of the poten TIACA TechnicalMemior&'nd No.; 9.71 tial function of the second iy "p .Adccording to (68a) and. (66), it is:  w(x,y) bnmEnm n) V m dn +r 2 d Pn1(Y d E + d Y 2m n4n(C) C os C 2 (69) The coefficients bnm are now so defined that wn is a function of. n only. The first term in (69) already depends on n only; hence the second term itself, which usually depends on E also, must be a function of T only. dni is a polynomial in Y with terms of the form d Y ynzp = [ cos c + n sin C] n2p hence is a sum of terms of the form (CosE)n2p1a (sin O) tnapa' Il (70) For the following arguments it is assumed that Mnm (c) is even in c ; so that all terms with odd powers of sin disappear in .the integration from 2 to +Z. But if a a2 2 is even, (sin c)a = 1 + sum of cos terms. When this is written in (70), Yn2p .consists of the following.sum mands: (cos e)n.aq n2pa 1a (70a) and the condition that n 2p a. n 2 q <. n 2 p Putting c = and = n + 1, respectively (" is an 2 2 4 '" % integer) and stipulating that 2 f (cos)nq bnmMnm(E) A C d ; m = 0; cos 2 (71) (72) (73) IACA Technical Memorandum No. 971 causes all terms containing the powers of cos C, i.e., po'ers of to disappear. The sole nondisappearing sum mand cf yr"p is obtained when n 2 = 0, that is, when first n is even and, according to (71) n 2p a = 0; this then reads, according to (70a), 7nn2p a.id depends no longer on c, so that it can be put before the integral. Combining the summands before the integral, which nov has the same value for every p, there is obtained conformably to d F1') in (69) for the integral d Y d Pn1(n) 1 i ,bnm E[m a / Cbn m n P A" d n 2 m Cnmos C (The integral disappears for odd n.) Mn" (C) contains the factor sin c; hence it is odd in r. Considerations corresponding to the foregoing then give the condition Cos Cn2q1 m _m/ d C (cos ) sin e bbnm n 1m) AEd = 0 = 1,2,..., (T 1); T= n and = n 1i (74) respectively 'he 7alue of the integral in (69) is other than zero only if n is odd. Summed up, it affords, by attention to 2,(K) = Eg+1() = 0 (equation (16)): 1 t r( ) 1 ( = i* Par 1i( )  War+i(TI) = k2r+i r r(T) = 2r V d V d n 1 d Par(n) d (,4) r d sr ('r,) 1 w.(ar+ 1)(T) = J2r+i ; War() = lar  dn d n whereby NACA Technical Memorandum No. 971 k2r+1 = b2r+1 mEr+i () +S 1ar = 1 /^ mr IMr (E) Le d c a (75) J2r+r =L sinE Z b;m+1 M2r+1(E)AE E 2 T m r r COS E m bnm and bnm satisfy equations (73) and (74), re spectively. They are a 1 homogeneous equations for the r, unknown bn and T 1 equations for the T unknown bnm, respectively, which can be determined therefrom up to a common constant factor. The latter is so chosen that k2r+1 = 'ar = /I K2; isr = Jnr+1 = 1 (75a) whence 1 (d \'%r(Tl) 1 \r(fl Pari(n) w(r+1f) (;2r ar()  .n a d T Lift. lift moments, and the lifting lines of the po tential function of the second type. These quantities are obtained by the application of the operator 1 cn cn1 dc to the corresponding quantities of the potential factor of the first type (42, 43, 44, 64). It is pointed out that, during the differentiation, the real content of the ellipse Fell emb'dies the factor c2. Then, bear in mind that (equations (68), (75), (75a)) gives the lift A = 8 V Tr 2 ell (77) Z"ACA Technical Memorandum No.. 971 O2 the pitching moment k= 8 aV Fll (8) *a the rolling moment L = c  V F (79) 3 2 i 2 1> E ;I 2K2 sin2E d C and for the lifting lines az(n) = 4 i Vc 1 K Po(n) 2 0 as(n) = 0 (80) a3(n) = 4 E V2 c l K2 P2(n) a4(ni) = 0 The second fundamental problem of airfoil theory. Thiz involves the calculation of the aerodynamic quanti ties of any given elliptic wing by means of the foregoing potential functions of the first and second types. The boundary condition to be met is given by x (x.y) = w(^xy)= dx (62a) 8 x V / d CO which, differentiated, leads to (46): w = V (46) cx z Posting as a linear combination of potential functions of the first and second types, leaves 7 = i anm n + n 'n +n + Dn n (81) n= m n n leaves on the disk (p = 1) "3,' m am nm na I 6 8 an m since . n 0 (81a) op= Z p=. az p= NACA Technical Memorandum No.:971 where then, however, a nm J En m() Enm() d d Fnm(p) (82) z c p v La( 4 ) P= Now, if is developed according to the Lame products n m n n n. ( m Em(p)= I2 Mnm() then, because of equations (46), (81a), and (82), based upon the orthogonality"of Lame's product, gnm = dFm()1 anm (83) Ld( p_ 1p=3 This defines the coefficients anm. of the potential func tion of the first type. On expanding z.= z( ,n) in a series of ( and n, the anm 's are determined by a comparison of coefficients in all terms affected with pow ers of  in the form of a screen method (reference 2). From equation (46) w S an dx= w(n) (84) V n,m o z V is now only a function of i. This residuary condition is complied with through a suitable linear combination of the downward function of the second type, which depends solelyon in. The condition reads n n ( )_ + Dn w_n() .= w(T) (85) n n This equation is integrated from n = 0 to n, and then multiplied by P2,i(,) and P2y(n), respectively  integrated again from n = 1 to +1, and so yields through the intermediary of the interrelations NACa Technical Memorandum No. 971 +1 d 2 2 a Pri ) P2,n) d 1 = 2(2ac1)+1 4cl  r o +1 / 2r(T,) P2 i(T) d T = if r = a if r a a(2a 1) r(2r+ i) two infinite systems of equations for the coefficients Cn and Dn  +1 / E C+r+ C 2 C / w(n')dn' P2a1(n)dn r=o a(2ol)r(2r+l) 41 / t o a = 1,2,... ,'/1 E i 2 Davy, 1 + 7 Dar = ~ ; / w (r,') dn'P2Y (n ) dn 47Y+1 r= 1 Y(2Y+l)r(2r1) V / / r( 1)d 'pav( .o 0 = 1,2,... The outflow condition. Apart from the compliance of the equation systems (86), the coefficients Cn and Dn must also satisfy the outflow condition, that is, they must be so chosen that the lift density disappears on the trailing edge. The potential function of the first type satisfies this condition without that, since they disap pear on the whole edge. The behavior of '*nm on the boundary is known with the application of the differentia tion process (65) to (41). [Qnm], c Mm() + ((./l^)) 1= f1 X2 a=. /c2 y2 (The term ((/ p2)) disappears with the root from the edge distance.) We post, respectively, Zbnm m() = () and bnm qnm(w) = Mn(P) (87) whence the potential function of the second type becomes =[n]p= bnm ) c Mn() + ((.1/ )) (88) P=1 Cy2  1 K Now the functions M n(c) have orthogonality proper ties similar to the trigonometric functions, as may be NACA Technical Memoran.dum. No. 971 proved by (7?3) and (74) if it is remembered that Mn(qP) can be written in the form An(c) = (sin p) C3[ao(cos )n31 + ag(cos )nC3 + ...] It is 3TT  ii1i(qp) Mn(cp) Acp dc = 0, if n = 2, 4,... (69) The outflow condition reads, according to (88): 7 Car Ilr(CP) + E0 C.r+i A2r+1(Co) 0 I<2cp< 3_L (90a) SDr M.Cr(C) + Z Dar+1 ?(ar+i)() 0 Di<%< L (90b) r=i r= i 2 which, after multiplication by Msa1(c) A 1) and I4s2y(cp)6Ac respectively, and integration from .= n T to 3T, give 2 2 Cg_11ia_1,_aM + Z Cr iarai = 0 a = 1,2,... (92a) D2V Ia.Vay + Z D^r+l I(ar+i) ,sY = 0 7 = 1,2,... (92b) r=1 whereby 3zT V/ liy(cp) 1i(CD) Acp dcp (93) The infinite equation systems (86) together with (92) then enable the determination of the coefficients of the potential functions Cn and Dn of the second type. These coefficients in conjunction with the previously com puted coefficients anm of the functions of the first type make the pr.ediction of the aerodynamic quantities of the given wing possible, as will be illustrated on a model case. Note. The calculation is made on the assumption that the flow strikes the elliptic disk parallel to the minor principal axis. The length of the principal axis in y direction was c, in x direction cl i3. The calcula tion is readily applicable to the case of elliptic wing in flow parallel to the major principal axis when K2 Is assumed negative, that is, supposing the aerodynamic cen ter on the y axis. The conversion formulas for the el liptic integrals with negative K2 are: ilHCA Technical Memorandum No. 971 25 n TI dc 1 ____ d _ J. I+( )sin2 c 2 J _C *o 1? S in > (94) / J1+ ( )sin2 dE =E /1K /l sin C dC './., 1K o *0 The integrals are as far as the factor before the integral, the same as for the reciprocal axes ratio. The flat ellintic disk in straight flow. Let the angle of attack be 0o, that is, the elliptic area is given by z = x tan ao = x Co Then we have, according to (62), 1 ( x, y) dw  = o; = 0 V dx on the disk. According to (83) therefore, the coefficients anm of the potential function of the first type are all zero; those of the second type must be computed conform ably to (73, 74, 75a), as exemplified here for n = 2 and n = 3. For an axis ratio of '(c)2 4 K2 KK = 2 = 0.96, that is, A = 6.37 5 Fell TT 2 the complete elliptic integral is: TT F = / d = 3.01611; Jo l 0.96 sin2 T E = / J1 0.96 sin2 dE = 1.05050 j/o In the case of n = 3 and E1 = 1; Cg = cu = 0 Es() = /1/TD (V2 Ps2) s = 1;2 M3 (E) = K2 sin2 p62 MS ( ) = 2 Ps NACA' Technical Memo'randum No. 971 and according to (20). it is: 1 1 3 = PS 2 Ps Ps 1 that is, pi = 0.19792; p32= 0.97008 Condition (73) and (75a) then read: 2 b33' / (0.96 sin2E 0.19792)LEdE + b3 IT (0.96 sin2 e 0.97008) A eC.d = 0  r= 1) l K2(0.960.19792)b, 3 1K (0.96 0.97008) whence (1) b3 = 1.317, = 0.3095 i. e., M3(cp) = 1.627 K2 sin2 c 0.561 For n = 3 and C = C2 = C3 = 1, we find E31() = 1 2 .. M 1_) 2 j; EX K M3K K Mv3i () = cos e sin e Equation (75a) reads: 1 d3 1 1j ,1 = . sin E b3 cos E sin CAC e = b / sin2 ACd Scos e / "' o that is, I = 2.654 b 37 = 2.654 Hence M._3 (p) = 2.E54 cos p sin Cp, /K 2 = K cos P (') The determination of the other necessary functions proceeds in similar manner. The integrals (93) can be com puted by means of the functions ILn(;), being either ellip tic or reducible to elementary. The coefficients ly 8 of IJACA Technical Memorandum No. 971 (92) are herewith known, Calculation of the right side of (86) yields +1 Ji 1 / ,nao0 for a= 1 ao I d n' P2a(n)dT=Co T1 for aa()= / 0 for a >1 1I c J For Y = 1,2,... the right side of (86) is always zero, whence no asymmetrical potential functions occur in y. In the effected calculation, the series (31) of the potential function was stopped with n = 4; hence eoua tions (92) and (86) must be taken for a = 1; t. It gives 2.101 Ci + 1.5217 C0 + 0.527 C4 = 0 (a = 1), S(92a) 0.2410 C2 + 0.5132 C + 0.9?24 04 = 0 (a = 2)J 1 2 1 2 C5 C + C2 c + c = 3 o (a = 1) I i 9 1 (3s) 1 C3 + 04 = 0 (a = 2) The direct solution gives C = 0.53 a0o, C_= 0.7785 ao, C3 = 0.342 cao, C4 =0.0135a0 and the lift, according to (77), at: S= 0.568 aox V Fell = 4.55 a.o Fell that is, d co The moment about the y axis is,according to (78): i, = G.7785 a x 0.9524 X c A/I K2 R 7 ?2  0 3 2 ell i = 1.98 ao c l Ka 2 ll the center of pressure is at x = 0.435, that is, cl :2 at 28.3 percent of the maximum wing chord. Incidental to the calculation of the induced drag, it is emphasized that the lift distribution of the lifting ITACA Technical 1'emorandnm No. 971 line does not disappear when:allowing only for an infinite number of series terms in (81) at the wing tips.(fig. 3), and hence must be included as a substitute lift distribu tion. In this case the elliptical is most suitable, giv ing for the induced drag the wellrknown formula a Fell Cwi = Ca .. For the axes ratio SK = = 0.75, A= 2.55 the procedure is the same, the quantities bnnm being as certained from (73, 74, 75a). This affords the functions Min(Cp) for the integrals ly 8, wherewith the coefficients of (92) are known. The solution gives 01 = 0.3741 ao, C2 = 0.6347 ,o, 03 = 0.2347 ?o, C4 = 0.0138 ao the lift being A = 2.99 ao P. V2 Fell and d a = 2.99 the pitching moment M = 1.397 ao 1 Fell and the center of pressure at x = 0 467 or 26.7 percent of maximum wing chord oYl K2 For I K 2 = 2' = 3 A = 0.637 it gives Ci = 0.124 ao, C2 =0.5245 mo, C3 = 0.066 mo C4 = 0.011 CLo that is, that is, d a = 0.99 d a. NACA Technical Memorandum No. 971 29 The center of pressure is situated at   = 0.584 = 20.8 percent of the chord For comparison the values for the flat circular disk are repeated (reference 2): Sd c = l.E2 S = 0, A = 1.372 d m.o center of pressure: z = 0.515, i.e., at 24.3 percent of c chori. T'e calculation method used here permits even a bound ary tr,.nsition to the lifting line (K' = .1 = 0). It affords, when two series terms are taken into account, d ca Sca = 2 n c.p. at 28.8 percent of chord. d ao The latter result corresponds to an elliptic spanwise load distribution with a center of pressure at 1/4 chord in each airfoil section (c.g. of a homogeneous semiellipse). Development of all quantities appearing in the equation systems vith respect to a affords for small K': d Ca 2 T d ao 1 63 i \ 4 7 1 ,2 1 + 3 K + ln 2 d c4 Then d ca is calculated according to linear wing theory d ao w'Lere, as is kno wrn, 1 d Ca 2 n A Aeff = 0o + Ca ' f.'. d do A + 2 there appears a marked discrepancy at small A with re spect to tnh ve.lues computed in accord with the theory of th5 lifting surface (fig. 4). On the other :and, the agreement with Jeinig's re sults is good. (See reference 6.) Figure 5 shows the center of pressure position plot ted against aspect ratio. The elliptic winz in yaw. This problem can be treated NACA Technical Memorandum No. 971 under the same assumptions as the wing in straight flow. The socalled angle of yaw B is defined in figure 6. row the streamlines are straights in z = 0, defined by y = (x xR)tan P + 7R = x tan 0 + const From equation (i) (V cos 0 + u) _ + (v sin P + v) 8 + 8x ay the plane (95) )w 1 P w a 8z p 82 under simplifying assumptions, there is found V cos V sin W 8 =V2 ax 6y d5 But because of (95) it becomes dw 8w 8w I= tan , dx ox dy that is dwcos P=wcos 0 sin w dx ax dy Hence eouation (96) can be written in the form cos V = V dx 8z (y = x tan P + const) The downwash follows from integration along a stream line again assumed as a straight line parallel to the di rection of flow w = 1 7 cos x S dx iH (97) The potential function .*. is again assumed as a linear combination of the potential function nanm, n, _n' in the form (81), whence the same forumlas are obtained for lift and moments. The downwash function is computed on the basis of an obliqueangle system of coordinates in the xy plane, givenaby the ellipse.diame.ter parallel .to the stream direc tion ( axis and the related.conjugate diameter  n axis. Posting (96) (96a) NACA Technical Memorandum No. 971 tan c: = ,/l 2 tan 5, tl/l 2K sin B that is, sin !3 = cos cos = 003 *.'p =  we have the quoted diameters given by tan c n y=  tan. O' +  and y = ,I respectively (99) 'l K On tiie disk, i.e., p = 1, it is according to equa tions (6) and (17): = cos y; cV/l K ./l nK Similarly, we post / 2 2 = 7 cos (, 0 ); that is, cos 6 /l K sin S  = z nTl Y S= = = L sin CP C r, = I sin (P q ) / sin (cos S^= 0 + (loo) In this instance the potential function nm con formable to (32) is utilized. During the respective dif ferentiations and integration along a streamline, the fact that y = x tan 8 + const should be borne in mind. Sm1 v'Y(t) e x m2 I e iea e3 c 1A (t) e,3 Y + L;e e e 3 'e /Y(t) Te z + nm(t) d t Vei eea e/ i e3  I (32) ,'ve find that a on(X) dCn(Xt)1 1e 82 d X c v/e ea /e e (98) 32 IACA Technical i4embrandum 0o. 971 and _ daQ(x) adQn(x) 1 E f (t) ea l'Y(t) e tan d x dX c L er/e2 e e3 eg e3 eg 8 n(x) cn(x) n3 thus can be expressed again by  ; posting 8 z dx the result in (97) and integrating over x gives wm 1 1 Y(t) eg wn 1 1  V 27Ti cos / e co2i c J sex eg/ex e3 x 1.. .. nm..(t) dt(101) JY(t) e Vy(t) e3 tan veg e lIe e3 /e3 e iez e With equations (51) and (52) Qn(X) is now replaced again by 1 F pn(Y(s)) dY Qn(x) M = I iT ds .2 X Y(s) as The integrand has poles at t = s, where X(t) Y(s)= 0 and at t = to, where I e "'2 t) e e (t) 3e tan = 0 sea eL/ea e3 e3 ei es ee i.e., 1Y(tg) e3 = i Ve7 e3 sin qc ; 7(tg) e3= e3 e3cos q The denominator at this point is as t t cos 9O i sin CP tan s + Li ie e2 e1 e J The residuum at t = t is therefore 11i.CA Technical Memorandum No. 971 2niQn ( (t )) nm (t6) i cos cU + i tan sin yp = 2n i B En ( Kcos ng) n (') (102) with a view to the fact that, according to (100) a(tp) = sin C. + cos Y = 1 Cl K and according to equation (33) Enm('S) = 1 Enm ) Sj eo= KCOS For the residuum at t = s, the same holds true as in the case of straight flow. It affords: ,r 1 iA c 2n i i P( cosc + n sin e) CS 2s 'iMn /n cos s yl"2 tan 0 sin E m( ) dE (103) By replacing 1l K2 tan "by tan Vc and posting the coordinates E and n by means of equation (100), the values (102) and (103) from (101) lead to : m )  / U CO[ S cos(c4p )+1n.sin(ra V)] Lc de (104) A 2 ./ cos(Z+, ,) For the terms of Pn' containing powers of g, the denominator cos(C + C%) in the integral disappears, af fording complete elliptic integrals of the first and sec ond types. To compute the others, numerator and denomina tor are expanded ,ith cos(c y)A1 P, which, with cos(c + og) cos(C ) A2, = cos2 A sin3 C gives (103) "NACA Technical Memorandum No. 971 Swnm ('.). = Enm(K.cOS cpP.)n(n() 1 + 2 nc s A 4A / Pnl pcos(^ )+ni(c+q)) (])Mnma) J_ cos21apsina (104a) 2 The existent integrals are reducible to complete ellip tic integrals of the first and second types and one integral of form 7 sin2 C dC r cos2 2 A p sinae Ac o which, as a complete elliptic integral of. the third type, is reducible to incomplete first and second types (refer ence 7.). 2 (lK2) sin A sin' C de_ cos / cos32A20 sin2c A 0 0 = E()P() E s F(0) A tan F( T (105) The calculation of the downwash function for 'the po tential function 1 is given as a model example. E() = IA a M (C) = 1 Pi(Y). = Qj(X) = In . 1 2 X 1 E' (K cos cp) =1 K cos =1 Then,according .to (1.04a).: .i.eCA Technical Memorandum No. 971 1 1 1 / 2 +n +H i 1 :, [i cos('+ n sin( cos + n cos d S / cos2cL'sin2C 0 Al( = 2. 2 ) S3 / T cos sin2 .he evaluation of the integral Lives w1 l L )2 p TT .'he downwash functions of the potential functions of the second type, According to (68) Sn = n n whereby ,nm 1 d [cn (m ,y,z,c)] cn' dc according to (65). The downwash function wn B',l) is computed from wn (I(,p) that is,  bnm Enm( Kcosy ) 1+ + 1( + Ed (106) 2 LP dY cos(c + (P) in the same manner. 36 ITACA Technical Memorandum Xo; 971 On the disk we have a p =0, that is,,on a streamline (i = const) the downwash function is constant; hence wn is a function of np only, and we accordingly put = 0 in d P (Y). To find the summand dY n1 n np Lsin (e + cp )]np from Yn2p n n 1 we put T = 7 and. respectively, whence 2 [sin (C + y )]n2p =[sin(E + p)]nETCsin( +p)]T2Tp =[sin(C + cpo)]n'2T(l + sum of cos terms) Factor cos(e + Cq) can be extracted from the sum,. thus becoming shorter with respect to the denominator. The new sum, however, multiplied by [sin(C + q)]1n2 yields only terms which either satisfy odd in C or else condi tions (73) and (74), respectively, and accordingly disappear. A proportion other than zero is afforded only by [sin(E + cp)]n2r x 1, which is, however, no longer depend ent on. exponent, p. In consequence, all n,,P before the integral can be combined conformably to d Pn() into d Pn (io ), since the integral for dY do every p is the same; i.e., Urn( T1) 1 a 1 dqniflp)  = E KbO1 CnP) o. 7 60 m d 7p + I s [sine ccosc + cos e sin  cos C cos r? + sin c sin cp n xMn(c) ede (107) cos2 As sin2 C Hence the downwash functions for the wing in yaw have the following form IL~CA Technical Memorandum. No. 971 V wn() d k(B) +dQ( in) n_() (107a) d np nTI 1 W'n() = ('n' d^ V^ ip( (10d_) w_ = () d .In.d + j n(P) d T(107b) The second fundamental problem for the case of the wine in sideslip. The procedure is the same as in the case of the wing in horizontal flow, by putting dw(x,y) V m ()E 7 1 ^uco s dw gn m .) m (n) (108) dx c n m where 1 w(.x,y) = cos z(x,y) (108a) V dx and y = x tan ) + const; \ to be given again by S= a m + Cn + Dn n (109) n m a n n n n ( Equation (96a) tien gives g 1 n I an (110) S.. .., a d(,p2 1) = and with it the coefficients of the potential functions of the first type. For the determination of the coefficienrts of the functions of the second type we take (97) in the following form: x x ; 'n dx + Zf dx + C + Q Dn [ ; cos Sn / 8z cos n n n cos i ,/no'm dx 1 w(x,y) a = w() (111) V n m 5z cos V The righthand side is a function of n only because of the determination of the anm from equation (96a). The rest of the formula then reads: I NACA Technical .KMeiorandum No. 971 n "n d Ti dnn (F) dn() ( + ) w( + n D In + nJn() W(J) (112) This equation is integrated from n = 0 to n and then multiplied by Pa (n) and 2y(.n) again from T = 1 to +1. This affords on the tasts of the orthogonality characteristics of the spherical functions 7 (Ckr+x aer+i( )+D2r+ 1 2r+i (P( )) 2a._l) 1r (2r +) + (C2ai2Ca() + Doaj2(P)) 2 4a 1 +1 TjP S / / w(n' )dn'pPa(np)dn; a = 1,2,... (!13a) 1 0 rx (Csr k2r () + D3 lar 12r 1) 1 + (C2+i1 is'+1 (P) + D2sg+, jsy+ij()) 1 I 4+ + I 1. f, (.1i. S/ w(Ti ') dn Pay(np) dna; Y = 1,2,.. (113) 1J / i " The outflow condition. The formulation of the outflow condition is prefaced by the following note. The downwash functions (107) onthe wing in sideslip have the quality of becoming infinite by n = 1 like Qn 3) and d Qn1 () respectively. But in = 1 are marginal d n , points of the elliptic disk in which parallels to the flow direction touch the ellipse (fig. 6). Accordingly, it is to be assumed that the socalled "vortex tails" in these points leave the disk parallel to the direction of the stream. Hence it is postulated that no flow around the trailing edge occurs between the points Ti = 1. In other words: lACA Technical Memorandum No. 971 SCnlin() + D3n n'() = 0 2 + < < + 3 This equation, multiplied by aX (p) A Lp and 2y(cp) LA, respectively and integrated from 1 + 0 to +3 + gives the fol 2 0 2 lowing systems of equations: Ca~I ac1,S1l2L+ S C2rI r, ai+ Z Dar.2r,2 20,1= 0; r=1 r= 1 a = 1,2,... +I WC (W0) ( 0 (114) D2rI_ ) 7, C2r12r_1,_2Y+ Z D2r+1U= r=yr1,Y (rr+1),Y V = 1,2,... 'Y/ whereby I IT = My (c) (iC ) Ac. dQ (114a) These two eatations ((113) and (114)) make it possible to determine the coefficients Cn and Dn. Together with the previously defined ann 's the aerodynamic quantities of an elliptic wing in yaw can be computed. As to the equation systems themselves, they form a coupling of the systems (82) and (92) for the case of straight flow, as is readily apparent from the similarity of the corresponding coefficients and which thus affords a first simple mathe matical check. 'The flat elliptic wing in yaw. The calculations are carried out for the axes ratio and the angles of yaw = 150 and P = 30 Again only the potential functions of the second type conformable to (108) and (110) are required. To simplify the voluminous paperwork only the functions up to the de gree n = 3 are taken, i.e., ITACA Technical Memorandum ITo. 971 i, a2 as, 3 (i .does not exist) The five unknown coefficients of these potential func tions require five equations. Two are taken from the con dition 1 w(n) = cos P tan so on the disk, equation (113a) for a = 1, and (ll3b) for Y = 1. For the other three the outflow condition (equation (114)) with a = 1.2 and Y = 1) is used. The Ln,(qc) are those previously de fined in the case of straight flow. The integrals 1) obtain now only the limits 1. + p and + 3 pT while TIY remain unchanged on account of the periodicity of the integrand. The values of the incomplete elliptic integrals of the first and second types necessary for the determination of the coefficients kn (), in(p), etc., were taken from Legendre's tables (reference 8) For p = 150 it affords C1 = 0.5204 a0 On = 0.7194 a0 De = 0.0144 ao C0 = 0.3343 co D1 = 0.0065 ao i.e. , A = 4.16 ao V. F 0 2 ell1 M = 1.83 a0 c 1 K2 2 V2 o a Tell 2 ell For P = 30 it is: 0C = 0.407 ao C2 = 0.562 .o D = 0.0277 a.0 C3 = 0.265 a0Q D3 = 0.0124 a i.e., A = 3.26 ao V2 Fell 2 M=1.43 a V2 e L = 0.074 a. cE a 2 ell NACA Technical Memorandum No. 971 The new additive moment L is positive according to the above calculations, which is synonymous with the fact that the leading wing half receives greater lift. The coordinates of the centers of pressure are: = 150: X = 0.440; c,/ K S= 30. x = 0.439; cl K2 TT  = 0.00925 = 0.0227 C (:p = 6002') (p = 13051,) For comparison we repeat the values obtained at P = 00 when the expansion is stopped uith n = 3. The bracketed terms contain the change in percent with respect to the quantities computed with the four expansion terms, and from which inferences can be made regarding the convergence. Ci = 0.563 ao (0.9 percent) C2 = 0.776 o.o (0.2 percent) C3 = 0.365 ao (+7.0 percent) A = 4.50 ao P V 2 o 2 ell 1 = 1.98 ,o cll K2 E V2 Fell 2 D2 = 0 D3 = 0 Center of pressure: = 0.4Z.8 c./JI K2 (+0.7 percent) The results are correlated in figures 7 and 8. For great A an approximate formula for the rolling moment in relation to angle of yaw and axes ratio K' =,/1 K2 is again expedient: dcL 15 d = ~ '; sin 2 do, 128 4 3 i tan + 2 In 1 2 sin n 243 1 +   K' cos 0 128 (115) The formula indicates that the moment on transition to the lifting line (K' = 0) disappears. But if the i.e. , NACA Technical Memorandum No. 971 moment with the half wing chordinstead ofhalf the span is made nondimensional, thus voiding the factor K' in' (115) the moment coefficient becomes logarithmically in finite on limiting transition. In the extreme case the lift decreases with cos2 0, for the reason that the flow velocity in x direction is V cooe . Horner's results on wings of different plan forms in sideslip are in very close agreement with the values given here. The assumption that the rolling moment is, aside from the angle of yaw, largely dependent upon the aspect ratio rather than the chord distribution appears therefore justified. Translation by J. Vanier, National Advisory Committee for Aeronautics. ' NACA Technical Memorandum No. 971 REFERENCES 1. Prandtl, L.: Beitrag zur Theorie der tragenden Flche. Z.f.a.M.M., vol.16, no. 6, Dec. 1936, pp. 36061. 2. Kinner, U.: Die kreisfirmire Tragflchie auf potential theoretischer Grundlage. Ing.Arch., vol. 8, 1937, pp. 4730. 3. Keine, E.: Handbuch der Kugelfunktionen. vol. I, Berlin, 18738, p. 3,17 ff; vol. II, 1881. 4. 4'hittakerWatson: Modern Analysis. Cambridge, 1920, p. 549 ff. 5. Hobson, E. W.: Spherical and Ellipsoidal Harmonics. Caibridge, 1931, p. 476. 6. Weinig, F.: Leitreg zur Tjeorie des Tragfligels endlicher, insbesondere kleiner Spannweite. Luftfahrtforschang, vol. 12, no. 12, Dec. 20, 1936, pp. 40509. 7. Schlomilch, D.: Comp. d. h6h. Analysis. vol. II, Braunschweig, 1865, pp. 33639. 8. Legenrdre, A. M.: Tafeln der ellipt. Normslintegrale. Stuttgart, 1931. 9. hoerner, S.: Kr5fte und riomente schragangestromter TrarflGel. Luftfahrtforschung, vol. 16, no. 4, April 20, 1939, pp. 17883. IACA Technical M.iemioreanium No. c'7 1 /C Z /* /C(, /K . * / /. , Figs. 1, ,3  i. '  r. ~i= wI  V __ 'I V*1., I ,  4. '~y. rrl L . ./ I   S ,. 0J r .11 j C,.9 .4 a J. ,'.r Svi Ki/.s L Conrutel lift jistriouti.n of the lifting. line. Elii.:,tic lift iistrib'uti.n wi.  e..al total lift. Fig. .3 W 4i.u., I 12 "jjJ 1i 1 32 L ~i1  1' II~/ II~1/ L C 4 I I u.2f L 1.0 'dIo i qi.1 1ase e 1 i^E ' NACA Technical iMemoranium n Io.71 ~711 I 4. C *C a 1 .5 'j1 U.4 0. 1 1. 1 L i rcu lar i i 'H: Liftin!l. sun..& *  Li f.i:, line  Tizo'irt 0. I U. .r 1.2 1.. Circul:,r Jii" Figure 5. Fils. 4,5 NACA Technical Mennraniurn ;o. 4I // /, / Figs.A, ", S V 91 / V cosB I/ i V',s i:,3 Sy TT  " Figure 6. Ic L drKx 0. 'i7" dc Figure 7 L ia for 1 .Ca daL 2 i U I .' :' u pl)tte! against angle of yaw p as.ect ratit /\ = ..?7 LC. 24 "0o plotted agist angle of ,'av 3 aspect ratioA\ = 6.37. Firure S. dc dfor for UNIVERSITY OF FLORIDA 1111 11 2 01 1 III I 1262 08106 312 4 