^A^nvTnTT^^B ». *•• i-'^*^** NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS TECHNICAL MEMORANDUM 1324 STEADY VIBRATIONS OF WING OF CIRCULAR PLAN FORM* and THEORY OF WING OF CIRCULAR PLAN FORM** By N. E. Kochin Translation **0b ustanovivshikhsya kolebaniyakh kryla krugovoi formy v plane.' Prikladnaya Matematika i Mekhanika, Vol. VI, 1942. ***'Teoriya kryia konechnogo razmakha krugovoi formy v plane." Prikladnaya Matematika i Mekhanika, Vol. IV, 1940. Washington January 1953 _^^,^ ^^ plORIDA ■ TON SClENUc udF^HT ""■'- nrPL 32611-7011 Uf GAINESVILLE, PL J^o r6 2- > 5 7 / NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS TECHNICAL MEMORANDUM 1324 STEADY VIBRATIONS OF WING OF CIRCULAR PLAN FORM* By N. E. Kochin The nonvortical motion of an ideal incompressible fluid has "been solved (reference l) for the case of uniform rectilinear motion of a wing of circular plan form. The method developed in reference 1 may also be generalized to the case of the nonsteady motion of such ving. The problem of the steady vibrations of a circular wing is solved herein. The results will be frequently referred to herein. The prob- lem of the steady vibrations of a circular wing was solved by another method by Th. Schade (reference 2). 1. Fundamental equations The wing^ the motion of which is under consideration, is assumed, as in reference 1, to be thin and slightly curved; its projection on the xy-plane has the shape of a circle ABCD of radius a with cen- ter at the origin of coordinates. The principal motion of the wing is assumed to be a rectilinear translational motion with constant velocity c parallel to the x-axis. The coordinate axes are assumed as displaced with the same velocity. On the principal motion of the wing is super- posed its additional harmonic vibration of frequency ui, where the pos- sibility of deformation of the wing is not excluded. The equation of the surface of the wing may then be represented in the form: z(x,y,t) = ^o(^^y) + ^i(xjy) cos cDt + ^2(^^y) ^^^ ^'^ (i-i) where the ratios ^^/^ ^^ well as the derivatives dL /Sx and 3L /Sy, where k = 0,1,2, are assumed small magnitudes. The fluid is assiomed incompressible and the motion is assumed non- vortical and occurring in the absence of external forces. The velocity *"0b ustanovivshikhsya kolebaniyakh kryla krugovoi formy v planel Prikladnaya Matematika i Mekhanika, Vol. VI, 1942, pp. 287-316. 2 MCA TM 1324 potential will be denoted by (p(x,y,z,t) and steady vibrations of the fluid will be assumed; that is, the velocity potential is represented in the form: cp(x,y,z,t) = (pQ(x,y,z) + (p^(x,y,z) cos oot + (f>^{x,y,z) sin cut It is evident that the functions (P„, iiU,y,o) = (1.11) in the entire xy -plane with the exception of the circle S and the half strip S on which (p^ iindergoes a discontinuity. The conditions (1.8), because of equation (l.lO), ass\ime the form: ^ = -53^ - kcpg = -g- + kcpi = on S (1.12) Finally, the absence of a disturbance of the fluid far ahead of the wing leads to the evident conditions at infinity: S\ ^\ S0 and satisfying the conditions: (H).^ - ^<-^> on S ^ ^ + ik$ =0 on H NACA TM 1324 (1 .18) • (1 .19) following from equations (1.9) and (l.l2). In the entire remaining part of the plane xy the following condition must be satisfied: 5>(x,y,0) = . (1.20) Moreover, the following conditions must be satisfied at infinity: (1.21) lim -^r— = lim -c— = lim -"t— = ^ -. ox oy oz x-^-H" x->-+» x-+- + <>» which are the boiaidary conditions of the first derivatives of the func- tion §(x,y,z) near the rear semicircumference BCD of the circle S and the condition that near the forward semicircumference DAB these derivatives may become infinite to the order of 6" / . 2. Fundamental formulas In reference 1 an expression was constructed, which depended on an ajrbitrary function fQ(x,y), which determined a harmonic fimction cp^(x,y,z) satisfying all the conditions imposed in the preceding section '^oi^}^,^) = 2^ J J fo(C.Ti) |K(x,y,z,C,Ti) /'X p- Jt 2 (2.1) G(x,y,z,r)Va^ - ^^ - 11^ cos r dr dx 1 1 (?^ + T]^ + a^ - 2a5 cos r - 2aT] sin r)J d 2 '' The functions K(x,y,z,^,T]) and G(x,y,z,y) for z>0 are given by K(x,y,z,5,Ti) =— arc tan Va^ - e^ _ Tj2 A/a2 - x^ - y^ - z^ +~ YT ar G(x,y,z,r) A/a^ - x^ - y^ - z^ + R (2.2) x^ + y^ + z^ + a^ - 2ax cos r - 2ay sin r MCA TM 1324 which are harmonic fimctions of x,y,z where r = a/(x - C)^ + (y - t)^ + z2 R = ^{a.^ - x^ - y^ - z2)2 + 4a2z2 In order to satisfy boiandary condition (1.9) (2.3) ^ = Zo(x,y) on S (2.4) it is necessary to take fQ(x,y) = - ZQ(x,y) + gQ(y) (2.5) where g^(y) is determined from a Fredholm integral equation of the second kind. The solution of the more general problem of steady vibrations may be presented in a similar form. Thus, f, (x^y) and f_(x,y) denote two arbitrary real functions, continuous, together with their partial derivatives of the first and second order, in the entire circle S; f^(x,y) + if2(x,y) = f(x,y) (2.6) It will now be shown that the function <^(x,y,z) = ^r- 2rt fU.n){ K(x,y,z,5,Tl) + 1_ e-ikx jt^VT \J+ atyj G(x,y,z,r) e ikx ^/ a - 4 - T] cos r dy dx 1 2 '^ {Kr + r\^ + ar - 2a5 cos r - 2ay sin r) ► d^ dTi (2.7) satisfies all the conditions of the preceding section except condition (1.18). MCA TM 1324 The f\inction G(x,y,z,r)> as shown by equation (2.2), is harmonicj hence the function L(x,y,z) = e-ikx/ e^^ G(x,y,z) dx ' + € will be a harmonic function. In fact, AT ^L SL SL SG .,_ -ikx/ ikx AL = — :r + — - + — - = ^5- - ikG + e -^^ e-^^^ dx2 Sy2 az2 ^x i r2 ::i„2 ay^ bz' dx 15 - ikG - e-ikx dx / eikx^^ + k^G^] dx When this expression is integrated by parts, it is easily shown that AL = 0, since both G and Sg/Sx approach zero for x ->■ <». It then follows that the function §(x,y,z) likewise satisfies the Laplace eqixation A"!- = (2.8) where from the form of equation (2.7) it is seen that $(x,y,z) is regular everywhere outside the circle S and half strip s. In exactly the same way it is shown that the conditions at infinity (l.2l) and condition (l.20) are satisfied. Furthermore , 3- + ik* = -;— ox Zn f(?,Ti) BK ^ + ikK + r> 2 « 1 n 2 G(x,y, z,r) A/a^ - g^ _ ^2 ^^^ ^ ^^ (^2 + ■r^2 + a2 - 2a? cos x - 2aT] sin y) dC dT] (2.9) NACA TM 1324 It is clear that if x^ + y^>a^ then » ^ + ik* = for z (2.10) so that condition (l.l9) is likewise satisfied. It thus remains to check the finiteness of the derivatives of the function (x,y,z) at the points of the semicircumference BCD of the circle S and to establish the behavior of these derivatives near the forward semlcirciomference DAB. But near the forward semi circumference ^ the inside integral in formula (2.7) evidently remains bounded^ as do its partial derivatives; since the first derivatives of the integral SI f(?,Ti) K(x,y,z,C,Ti) dK dTi as established in reference 1, and as will again be proven, have near the contour of the circle S the order b'-^/^ (where 5 is the dis- tance of a point to the contour ABCD of the circle s) , it is clear that the first derivatives of the fiuiction ^(x,y, z) also have the order &-1/2 near the forward semicircumference DAB of the circle S. For determining the behavior of the function ^(xjy^z) near the rear semicirc\imference BCD, the right side of equation (2.9) is trans- formed. Denoting it by M(x,y,z) and making use of formula (2.1l) of reference 1 and the formula of integration by parts (2.14) of reference 1, M(x,y,z) 2-n J J f(d), ikK - ^1^ "2 " G(x,y,z,r) ya -€ -T] cos r dr j (4 +T1 +a -2a5 cos y-Zar] sin y) 'd? ^n T-. ^iT II K d? dn (2.11) 10 NACA TM 1324 It is evident that this function remains finite near the rear semi circumference BCD. But when the following equation is integrated, g+ ik = M(x,y,z) (2.12) there is obtained (x,y,z) = e-i^l e^^ M(x,y,z) dx + #(0,y,z) e-^^ (2.13) whence it is clear that both the function $ and its derivative with respect to x remain finite near the rear semicirciimference BCD. The derivatives of M with respect to y and z will be of the order &"V^ near BCD, as follows from a consideration analogous to that which was adduced previously for determining the behavior of the function«-<|> (x,y,z) near the forward edge of the wing DAB. Since r^x e^ ikx |M ^ ^ S'l>(0.y,z) g_iiuj oy dy it is clear that the derivative b^/hy , and similarly 5l>/3z, remain finite near the rear edge of the wing BCD. The function (2.7) thus satisfies all the imposed conditons. The only condition not utilized was condition (l.lS) = Z(x,y) on S (2.14) z=0 When the following formulas are employed: NACA TM 1324 11 11m z -♦+0 J J ^ ^^^'"^^ dC dTi = - 27tf(x,y) lim Z-+.+0 ^v a^ - x^ - y^ - z^ + R = aA^ y;2 , ,.2 „2 for x2 + y2 < a2 for x^ + y^ > a^ X" + y^ - a'- it is found without difficulty that on S ■ikx {tLo - - ^<^'^' ^ «'^> (2.15) where ■ gCy) 2^ n* -y r.2 " ^^ +- ^ gikx (^2 ^ y2 . a2)-l/2 (a^ - g^ - n^)^/^ cos r f(g,l) dy dx dg da_ (x' + y*^ + a^ - 2ax cos r - 2ay sin r)(€ + i)' + a'^ - 2a€ cos r - 2aii sin r) (2.16) The following equation is thus obtained: -f(x,y) + g(y) e-i^ = Z(x,y) whence f(x,y) = - Z(x,y) + g(y) e'ikx (2.17) (2.18) Substitution of this value of the function f(x,y) in equa- tion (2.16) yields, for the determination of the function g(y), an integral equation of Fredholm la g(y) = W(y) + / H(y,Ti) g(Tj) dTi (2.19) where N(y) = - -S ..223 gikx V^2 . g2 . ^2 G(x,y,z,r) cos r Z(g,Ti) dr dx d6 dT) ^x^ + y^ - a^(5^ + H^ + a^ - 2a? cob x - 2aii sin r) (2.20) 12 MCA TM 1324 with G(x,y,z,r) according to equations (2.2) and 2«3 a2-y2 f^U^ 3 \(.^-y^ HCy.n) elk (x - 0(^2 ^ y2 . ^2)-V2 (a2 . g^ ■ ^2)^ cos r dr dx dC (x^ + y2 ^. ^2 . 2ax cos r - 2ay sin r)(C + 1 + a^ - 2a? cos r - 2aii sin r) (2.21) 1 2 " 3. Conrputation of forces The press\ire p may be determined from formula (1.6), which with the notation (1.14) may be written in the following form: (; p = pc S-T-^ + Re ^0 + ik#) e' •icot (3.1) For the computation of the forces acting on the wing, it is neces- sary to know the pressure on the circle S. Because of equation (l.lO), the pressures above and below the wing differ only in sign: P_ = - Pj (3.2) For clarity, the signs of the functions on the wing will henceforth be ass\jmed to be the limiting values in approaching the wing from above, that is, for z-*+ 0. For the lift force P the following expression is obtained P = JJ (p_ - p+) dx dy = - 2 \ \ p^ S dx dy = 2pc in ^0 „ 'c — + Re ox (i * ^^) ■ioot * dx dy (3.3) NACA TM 1324 13 But ty formulas (2.9) and (2.1l), the following equation applies on the uDDer side of the circle S: Mix^ + i lc-+ yields the formula / r K(x,y,0,C,Ti) d? dT] = 4 tsJB^ - x2 - y2 on S which is equivalent to equation (3.6), since K(x,y,0,^,Ti) is a symmet- rical function with respect to the points M(x,y) and W(^,r|). The -following formula is thus obtained: \~^^^ - 1 k*(x,y,0)1 dx dy = I I L/JT^r-^2 fg , ,^1 d? dn - 1 ,2 " Va2 - ?2 . ,2 f(5,^) / , '°' ^ -^^ — d€ dn (3.10) . ^ ?^ + ri^ + a^ - 2a ? cos r - 2ar) sin r MCA TM 1324 15 If this ejqjression and similar expressions are substituted for the function (Pq, obtained from equation (3.10) for k = 0, the final expression of the lift force acting on the wing is obtained: P = 4pc' A/a2 - e - r,2 go . Re i 5f0 -icDt (I ^ ") ^[f0 + Re(e-io3tJf] cos y dy 1 „ K"^ + vfi + B? - 2a5 cos r - 2aT) sin y" } dC dT] (3.11) By integration by parts and with the aid of the following formula 2 jt J ^2 + Ti2 + cos y dy 2nK a^ 2a4 cos y - 2ai] sin y a(a2 - ^2 _ ■qE) (3.12) equation (3.1l) may be rewritten in the form: -// Re(ikfe-i^^) + 3 . 2 |L [fo . He(.e— ]J^ ^-^ cos y dy a2 - 2a4 cos y - 2aTi sin in y r d? dTl (3.13) In a similar manner, the formulas for the moments of the forces about the x- and y-axes are obtained. For the moment of the pressure forces about the x-axis -ss- M^ = / I y(p- - p+) dx dy = - 2 / / yp+ dx dy (3.14) SP 16 MCA TM 1324 there is obtained Mjj. = - 2pc r^yJ^ + Re /"I* + i k *) e-i"^^ Idx dy (3.15) The order of integration is interchanged by use of equation (3.4) It is here necessary to compute two integrals. By formula (4.44) of reference 1, >Va^ - x^ - y^ dx dy 4 o n j e-i"^^ »dx dy (3.25) It is here necessary to employ the formulas fS': xA/a^ - x^ - y^ 2 + y2 + a2 - 2ax cos y - 2ay sin y dx dy = 4 2 — na'^ cos r 3 (3.26) NACA TM 1324 19 / /x K(x,y,0,?,ii) dx dy = I cV^^ - C^ " n' (3.27) As before, there is obtained M^ = 8pc y " 3: aP^ 2 - n2 ^4 Bf, ^ ° + Re r2 „2 / 3 a^ - C 2 fo - R- e-icDt h f _ ii,^ fj 3 n I, [fo . Re(e— ,)|^ -^-^ cos y dy a - 2a5 cos y - 2aT] sin yi d? dT] (3.30) The value can now be computed for the frontal resistance W, which is composed of two parts. First, the normal force (p_ - p^) dx dy acting on an element of the wing dx dy will have a component in the direction of the x-axis: 20 NACA TM 1324 bz (P- - P+) ^ cbc dy = (p. ^ - «Hi ^"'"')' ^^ ^^ if ;(x,y,t) = ^o(^'y) + Re[c(x.y) e-io^j is the equation of the siirface of the wing. Integration of this expression gives the first part of the frontal resistance in the form: 2pc ^ + Refl^ e-io^tl ox Vox •) :>-iXJClt dx dy 'c — + Rel-^ e ox \dx icDt 1 dx dy (3.31) In fact, the frontal resistance W will be less than W, , since a suction force Wo appears because of the presence of the sharp leading edge of the wing DAB; therefore. W Wi - W2 (3.32) The suction force Wg is connected with the presence of a strong rarefaction near the edge of the wing. This rarefaction is tsLfcen into account principally by the square terms of the fundamental formulas (1.3) or (l.S) for the pressure and it is therefore unnecessary to employ these fonaulas here. The suction force Wg is computed from the law of conservation of momentum applied to a thin filament-like close region t containing the forwaxd semicircumference DAB of the circle Sj region t is bounded outside by siirface a and inside by part S' of the upper side of circle S adjacent to the semicircumference DAB and the part S'' of the lower side of the circle S. Figure 1 shows a section of these surfaces obtained by a passing plane through the z-axis. The equation expressing the momentum law is projected on the x-axis: - Wg - / / p cos(n,x) dS=/ / /p^dT + p / / v^v^^ dS + p / / ' Vn^x dS (3.33) S'+S' NACA TM 1324 21 The left-hand side is the sum of the projections on the x-axis of all the forces acting on the volume of fluid considered^ and on the right-hand side is the total derivative with respect to time of the component on the x-axis of the momentum of this volumej this derivative consists of two parts, a volume integral connected with the local change of velocity and a surface integral expressing the transfer of the momen- tum of the particles of the fluid through the bounding siorfaces of the volume T . Equation (3.33) may be written both for the stationary system of coordinates Oi^iyiZ]_ and for the moving system of coordinates Oxyz. For the stationary system of coordinates, expression (1.3) is used for the quantity p; moreover, Vy = 'n (3.34) By the theorem of Gauss dr = cos(n,x) dS + dT S'+S" p -sr- cos(n,x) dS (3.35) From equation (l.3) and the equation just derived, the following expression is obtained from equation (3.33) after a number of simple transformations : Wo = -^ cos(n,x) dS cos(n,x) dS - l^l^ds ox on (3.36) Since Scp/Stj^ and Scp/Sx near the leading edge of the wing are of the order 6"V2 and S

^) ^a2 - g2 - ^2 d€ dT^ ^3_^^. ^2 + T]2 + a2' - 2a4 cos - 2aTi sin For this purpose, the following difference is estimated: A = J^(x,y,z) - N(0) The circle S is divided into two parts: the circle S-^ of radius a - e; and the ring Sg lying between the circumferences of radii a - e and a. ^1 = / ^(^'^^ Va2 - ^2 . ^2 ; 2a ^ 1 U ^ i^ J J |2a^r2 + (a^ - K^ - t\^){b.^ - x^ - y^ - z^ + r) Tq^ / / ^ -^^2 ? JZ^Tr?- - Zbl^t^ - (a^ - ?2 - n^)(a2 - x2 - y2 - z2 + R) •(5,n) A/a'^ - ?^ - T)-^ :; Z '-12 1 ' d? dii Si ^2a2r2 + (e.^ - ^2 - ,i2)(a2 . ^2 . y2 . ^2 ^ rJJ r_2 ]r,^2y.2 NACA TM 1324 25 ro^ - r^ = 5^ + Ti^ + a^ - 2aC cos 6 - SaT] sin 6 - (x-^)^ - (y-T))^ - z^ = 26 cos a(4 cos + t] sin - a) - S^ 2a2(ro2 - r2) - {e^ - ^^ - y{^){e.'^ - x2 - y2 _ z2 + r) = - 2a5rQ2 cos a - (a2 + ?2 ^ ^2) g2 _ ^^2 _ ^2 _ ^2) j^ Since Tq^ < 4a2 R ^ 2a5 + &2 therefore |2a2(ro2 - r2) - (a2 - K^ - r\^){a.^ _ x2 - y2 _ z2 + r) | ^ 2a5rQ2 + 2a2s2 + (a2 - ^2 _ T^2) ^ Hence if |f(C,Ti)| < M in the circle S then |Ai|^2a5M' ' 'Va2 - |2__ ,2 ^^ ^^ 2a2r2 + (a2 - ^2 _ T]2)(a2 - x2 - y2 - z2 + R) 2a262M Va2 - ^2 _ ^2 ^. ^^ ro2[[2a2r2 + (a2 - ^2 _ T]2)(a2 - x2 - y2 _ z2 + r)] RM V(a2 - g2 . ^2)5 ^g an ^ ro2 [^2a2r2 + (a2 - ^2 _ T^2)(g^2 _ ^2 _ ^2 _ ^2 + r)J ^l' But by equation (2.24) of reference 1 'Va2 - g2 _ ^2 ag ^^ ^ ^ _ 2a2r2 + (a2 - ^2 _ 7^2) (g^2 _ x2 - y2 - z2 + r) a o Since 2a2r2 + (a2 - ^2 _ Ti2)(a2 - x2 - y2 - z2 + r) = 2a2rQ2 + 2a5ro2 cos a + (a2 + ^2 ^ ^2) ^2 ^ (g^2 _ £;2 _ ,^2^ j^ 26 MCA TM 1324 hence for 5 a^rQ^ (3.46) and ^2^a.2-g2_T^2 ^^ ^^ 'Va^ ^2 1^2 ^p ^- r 2r2a2r2+(a2-^2_T^2)(a2-x2-y2-z2+R)] ^ J / tq^ + p'^)'^ J a- 6 V(a2 - p2)5 p dp d^ _ ^ 2n(a2 + p2) p dp ^(a2 - p2)3 -.p=a-e V + 2it^i/2ae - e'' - 6jta< 2a6 - e2 2^^ y\fr As a result, the following inequality is obtained U I < o ,,c 8jtM52^\/~S 47:M(2a& + 82) lAil $ 2rtMS + — ^ + ;— ^ MCA TM 1324 27 The difference is estimated ha 2a2 f(g,Ti)-Va2 - g^ - n^ dg dr, _ / / ^ , . ^^2 . £2 . .2 dr dn 2a2r2 + (a2 - ^2 . ,,2)(a2 . x2 - y2 - z2 + r) J J tqZ ^^'^' ^^ ^ '^ "^^ '^'^ On account of equation (3.46) ^2 [Agl^ 3M / / ^ "Va^ - ^2 _ n^ d? dTi But y / ro2 V / I p2 - 2ap cos* . s? I ^,2 - p2 and therefore I A2|< 6jtM'\/2a6 Thus for A = A-]_ + Ag the estimate is obtained Assuming e = 6 yields I A|< 24nM/\/a5 Thus 2a^A/a^ - g^ - Ti^ f(g,T|) dC dTi ^ 2a2r2 + (a2 - ^2 _ T]2)(a2 - x2 - y2 - z2 + r) ^ ^ (^) ^ (3.47) 28 NACA TM 1324 where 0(a) denotes a magnitude, whose ratio to a remains finite when 6 approaches zero. An estimate of the second integral entering equation (3.40) is given:. J2(x.y,z) = gf(g,T]) d^ dT] ^a.2-^2.^2 |2a2r2+(a2-^2_i^2) (a2_x2_y2_z2+R)] (3.48) Again assuming 6 < a/2 yields |J2I^! d^ dTi p dp d ^ ^ ' ; Va2 - ?2 . ^2 (^^2 ^ 52) 2 OvVO 'Wa^ - p^ (a^ - 2ap cos * + p^ + 6^) 2m P dp 'wa2 - p2'W(a2 + p2 + 62)2 . 4a2p2 but a-e a-e P dp P dp V^2772 Y(a2 + p2 + b2)2 . 4^2p2 / ^(^^2 _ p2) Va^^^ p=a-e p=0 P dp 1< 1 Y2ae - e2 ^ V^ P dp '\/a2 - p2/i/(a2 + p2 + §2)2 _ 4a2p2 / 2a.b^a? - p2 ^^jyr^.2 a^ - p' 2a& p=a P=a-e 2a& 6'\/2S NACA TM 1324 29 hence and for 6=5 From equation (3.40) and equation (3.42)^ the following is obtained on account of the estimates (3.47) and (3.49): ox naR ^ ' ^ ' In exactly the same way, there is obtained |y_ VrN(a)yVa^-x^-y^-z2. _J^^(^^ (3.5l) oy jtaR ^ ' ^ ' Finally, But |S = - _^ arc tan A + 2A_ [ ^ ^ z(a2 + x^ + y ^ + z^ - r)1 oz Jtr3 jt(l + a2) I r3 rR(a2 - x2 - y2 _ z2 + r) where ^/2 72 2'\r2 2 2 2 ~ A = V a - g - T] ya -X -y -z +R arV2" Assuming z > 0, r' ^ dC dri •$ Zn .0 30 NACA TM 1324 hence ^^ arc tan A+ ^ ^ .^ -^ f (d) d? dT] Ljtr- n 1 + a2 rSy ^ 2(jr + l) M and therefore SU ^ 2V2"az(a'' + x'^ + y^ + z'^ - R) ^ f(e,n)^a2 - £^ - n^ dg dn + 0(1) Again use is made of equations (3.47) and (3.42) and the fact that for z>0 without difficulty: .iV^.3.„a.„3.,. a + X + y + z SU a^ + x^ + y^ + z^ - R ^w^na/7 2 2 2 2 ^/-, ^ ^ = N(0) \/R - a^ + x"^ + y^ + z^ + 0(1) itRa^Y^" ^ (3.52) From what has been said previously about equation (3.38) it is evident that if F(^,Ti,t) = fo(?,Ti) + fi(^,Ti) cos oat + f^iK^n) sin oat (3.53) N(0,t) = FiK,^,t)^|a.^ - g2 _ ^2 ^^ ^^ ^2 + Ti2 + a2 - 2a? cos 9 - 2aT) sin 9 (3.54) the following results NACA TM 1324 31 xA/a2 - X^ - y2 _ 2,2 + R Vs'it^aR yVa^ - x2 _ y2 _ 2^ + R W(0,t) + 0(l) N(0,t) + 0(1) (3.55) 1 ^ ~ 2V^ n2^2R (a^ + x^ + y^ + z^ - r) X ^\/r - a^ + x^ + y^ + z^ N(e,t) + 0(l) or^ in the coordinates &, 0, a ^_, N(0,t) cos sinf-;; a) I? = i V^_Z + 0(1) ox ^(p N(0,t) sin sinfg al ^ = it2/^2a& ^(p W(0,t) cos(-| a) + 0(1) ^ Jt^ 'Y2a& + 0(1) (3.56) The computation of the suction force W„ by equation (3.3?) is considered. An arc D'AB' of the circumference C is taken synmetricELl with respect to the x-axis with subtending angle Z6q<-!X. For the sur- face a, the part Oq is taken of the surface determined by equa- tions (3.4l) for constant bg, where changes from -6q to + 0q and a from -n to +jr and two bases, one of which, a-^, corresponds to = 0Q and the other, Og, corresponds to = -0q, where on these bases & varies from ■ to 5q and a from -n to +jt. On the toroidal surface: cos(n,x) = cos a cos cos(n,y) = cos a sin 9 cos(n,z) = sin a N(0,t) sin(i a) ^ " "Sx '^°^'^^>^) + ^ cos(n,y) + -^ cos(n,z} \2 .2y + 0(1) 2a5 32 MCA TM 1324 Hence simple computation shows that cos(n,x) dS - / / g g dS 1 I I -2— N^(e,t) cos a cos e de da + 2 ' ' ' J-6q <-/-" y-00 '^-'' '00^ -L. N^fe.t) cos e sin^ i a de da + 2«4 ' ' ' 2 .^0 o(Vso) = "S / N^(S't) cos e de + o('>/6j) 2if ;-er In the same manner, the integrals taken over the bases a-^ and ag have the order 0(&q). Hence if 6q approaches zero, for the suction force developed along the arc D'AB', the following expression is obtained ^0 2jt3 N^(0,t) cos e de Now when 6q approaches rt/2, the required expression for the suction force Wg is obtained in the following form: Wp = 2rt3 N2(e,t) cos 6 de (3.57) i" The mean value of the frontal resistance is found. Equation (3.3l) shows that for the mean value of W^ Wi = 2pc ScPo 5^0 1 /^ (4.14) Integration of equations (4.12) yields 38 MCA TM 1324 ^o(X'y) = T ^ " c ^^^^ ^ ^ ^^^^ (4.15) where hQ(y) and h(y) are arbitrary functions of y. The function go(y) was obtained in reference 1, where, however, errors slipped into the computations. Setting y = - a cos Hq(0) = -^ sin ggC- a cos 0) (0< 0< n) (4.16) gives in place of equation (4.22) of reference 1 1 - sin -p -. 1 - cos -^ cos — In ^ — + sin ^ In ^ — (4.17) 1 + sin -^0 1 + cos — Hence setting hQ(y) = and Aq = ac in place of equation (4.23) of reference 1 yields f t \ J 3 1 L A/2a + A/a + y r 1 L Vsa + ^Va - yV ^o(x,y) = ax< J - — 5 In ^f— : ^. 1 - -^ In -^=: '', A^- A/aTT/ S^^ \ V2^- -ya - y ^\/2a VZsT - ^a + y ^\fZa V2a - >\/a - y 2jt2 -^a + y A^a* + ^a + y Sn^ y^a - y '\/2a + /\/a + y '(4.18) In particular for y = and y = + a/2 the following values are obtained in place of those given in reference 1: MCA TM 1324 ^(x,0) = ax 1 - jz ln2(V2' + l) + ^ ln(V^+ 1) =» 0. <-!) = ax ^ 2n^ ^Y3) --i^ ln23 + ^^ ln(2+ V3) + ^ In 3 39 ' 0.9146 ox In the same way, the expansion given in reference 1 of the fiinction ) in a trigonometric £ replaced by the following: H„(0) in a trigonometric series in the interval 0^9$n should be TT /n\ • aA^ V^ Y^ sin(2k + l) 6 (1 1 1 \ Ho(e) = sxn eb - 4) +2^ k(k I l) (3^5+- • • + 4k^ '^ k=l ^ that is, (4.19) Hn(e) P2k+1 sin (2k + l) where P-L = 0.9348 P3 = 0.2667 P5 = 0.1312 Py = 0.0796 P9 0.0504 In connection with this, corrections should also be applied to the numerical values, which are given in reference 1, of the coef- ficients Bq of the trigonometric series for the circulation obtained by the usual theory B-^ = 2.2125 aca B5 = -0.0296 aca Bg = -0.0067 aca B3 = -0.0934 aca By = -0.0133 aca Hence for the lift force in place of equation (4.29) of refer- ence 1, the following is obtained: Pq = 2 npca B-|_ = 3.4755 pc^a^a 40 NACA TM 1324 which exceeds the accurate value by 36 percent. For the induced drag, in place of equation (4.30) of reference 1, the following is obtained Wq » 1.9350 pc^a^a^ which exceeds the accurate value by 87 percent. Corrections are made in the third example given in reference 1. The value of the definite integral is: >1 ^^L^dy = ^-iln2(V^+l) Hence in equation (4.52) of reference 1 the coefficient of sin Q cos 9 is simplified and assumes the value -5n^/^. In equa- tion (4.53) the coefficient of sin 20 was incorrectly computed} its correct value is 5p = - -^ + 4r = -0.14555 ^ 8 9 In this connection, the value of the coefficient B2 should also be corrected: Bg = -0.7436 aca^ For the induced drag and the moment of the forces about the x-axis, in place of the values of equation (4.55) of reference 1, the following is obtained: W = 0.4343 pa^c^a"^ M^^. = 0.5840 pa^c^a^ the first gives an error of 140 percent; the second of 55 percent. The shape of the wing obtained An 1 Z(x,y,t) = — X - - gQ(y) X + Re< e-^^ A iB\ 1 - e-ikx iBx 1 / % -ikx > f, on^ A ^ X — ik^ — kT - c s^y) ^^ 1/ (^-20) I depends on the frequency of the vibrations and is deformed during the vibrations. The rigid wing is of greater interest. It is possible with the aid of the results obtained to obtain an approximate solution of the problem of the vibrations of a plane cir- cular wing for small frequencies of vibration. NACA TM 1324 41 The case is now considered of a wing varying its angle of attack periodically according to the harmonic law (4.1), so that equation (4.2) holds. If fo(x,y) = Aq f (x,y) = A + Bx eq-uation (4.2) yields Zo(x,y) = - Aq + g(y) Z(x,y) = - A - Bx + g(y) e''^^ (4.2l) If Aya2ly2 ^ ^ / 2 Go(y) = ^ ^^ ^ "°" ^ ^^ ^ jt"^ ' + - . Ig n / \jx^^ + y^ - a^ (x2+ y2 + a2 - 2ax cos y - 2ay sin y) Va2-y2 £ rt 3 2 / •, _ a£ / / eil"^ cos y dy dx i^^ / / 'Wx2+ y2 - a2 (x2 + y2 + a2 - 2ax cos y - 2ay sin y) 'va^-y^ — jr G (y) = .§^ / / ei^ cos^ y dy dx Ti^ I I A /x2 + y2 -a2 (x2 + y2 + a2 - 2ax cos y - 2ay sin y) 2 (4.22) Then go(y) = AoGo(y) g(y) = AGi(y) + BG2(y) (4.23) In place of G}^(y), their mean values are taken over the area of the wing: Gt = I / Gk(y) Ya2 - y2 dyi J / -\/ aZ " y^ dyl = -^ / C5k(y) ^J e.Z - y2 dy (k = 0,1,2) (4.24) 42 MCA TM 1324 If the frequency of the vibrations is ass\amed small, or more accurately, the magnitude ka is ass\imed small, the expansion e-ikx = 1 _ ikx - i k2x2 - . . . 2 may be limited to the first two terms. From equation (4.2l), the following approximate expressions were obtained Zo(x,y) - - Aq + AqGq Z(x,y) « - A - Bx + (1 - ikx)(AG^ + BGg) Comparison with equation (4.2) results in: - cpo = - Aq + AqGo - c3-i_ = - A + AG-L + B'Gg - cP-Lik = - B - ik(AG^ + Bo'g) (4.25) whence cpn cPi (1 + 2ikGp) cp-,ik(l - 2(h) Aq = ^ A= — ^^-^ =F-. B= — ^^-^= £r (4.26) 1 - Gq 1 - Gq^ + ikGg 1 - G-L + ikGg The following is con^iuted a Gq = -^ / Go(y) Va2 - y2 dy = I / Go(- a cos e)sin2 9 de d-a Jo But by equation (4.16) gQ(- a cos e) 2. sin Q Gn(- a cos 0) = sin 9 — '■ -. = — ^ Hn(0) hence, expansion (4.19) is used, yielding MCA TM 1324 43 Go = ^ /■ Ho(e) sine d9^^(4 - ^ "" ^ ^ V ■0 - ^ / -OV^^ --- ^ "" = ;;3 \^-2- - 7 2 = 2 " ^ and therefore Go = I -0.4053 = 0.0947 Aq = I.IOScPq (4.27) Equations (4.26) show that in coniputing G-, it is stiff icient to use the terms of first- order smallness relative to ka, while in com- puting Gg it is sufficient to use the principal term not depending on k. For small ka the following res\ilts Gi = Go + ikGii + ofk^a^ In ^J G2 = G20 + 0(ka2) (4.28) where G-^ and Ggo sire the mean values over the area of the circle S of the functions ^\f7^::^ ^ a'-'y'- -X It Wx^ + y^ - a^ (x^ + y^ + a<^ - 2ax cos ■f - 2ay sin r) \^ G20(y) =14/" r , '^"^^ ^ ^^ '^^ (4.30) / a/^^ + y2 - a2 (x2 + y2 + a2 - 2ax cos "if - 2ay sin x) 3 „2 ^+ - O 2 " In fact; Gi - Go - ikG-Li = ^ / / / G*(x.y.r) dr dx dy -y2 3 „ .2 |-| « (4.31) where G*(x,y,r) = (ei^ - 1 - ikx) cos rVa^ - y2 /^x^ + y2 _ a^ (x^ + y^ + a^ - 2ax cos r - 2ay sin r) 44 MCA TM 1324 The interval of integration with respect to x is divided into two peo-ts: from'W s? - y^ to 2a and from 2a to «. Since for a>0 |eia - 1 - ia|< a2 in the therefore interval A/a^ - y^2a, I y| < a, n/2^y<3it/2 the inequal- ity holds ? ? ? 3 ? x^ + y"^ - a^>- x^ (x - a cos y) + (y - a sin t)^ x As J '^ cos Y" dy = -2 a '1 '2 " 2 dy = — ^ e^^ - 1 - ikx = cos kx - 1 + i(sin kx - kx) the following inequalities are obtained when, for clarity, ka is assumed ^1, a 2a "5 Jt G*(x,y,r) dr dx dy < 4a2 J / 1 - cos kx (ix + J kx - sin kx VTnsl / x3 / x3 ^ ^2a J 2a dx> 4a^k' 2^2 1 - cos u 4a^ A 2 du + u - sm u du 2u /2ak du + /2ak 2 , n JO = 2^2 4a''k'' J 1 , 1 1 V. < 0.25a2k2 + 0.12a2k2 In ^ (4.33) NACA TM 1324 45 Combining inequalities (4.32) and (4.33) yields, on account of equation (4.3l), I G^L - C Q - ikG-L-L|< O-SSa^k^ + 0.12a2k2 In ■— ak from which the first of the estimates (4.30) follows. In BJi entirely analogous manner, since, for a>0 |eic^ - l| (4.42) The fluctuation in the lift force due to the vibrations of the wing thus leads the latter in phase, the maximum value of the lift force being greater than the value which was obtained in the computa- tion for the steady motion. In the same way, eqiiation (4.6) leads to the following expression for the moment of the pressure forces about the y-axis: 1% = - pc^a^ <1.473po + Pi(l.473 cos cut + 0.867 ka sin cot )> (4.43) The component of the frontal resistance Wj^ i^ determined in the given case by the evident formiila W2_ = P(Po + Pi cos tut) that is, W-L = pc^a^ <2.813Pq^ + 1.406P-l^ + PqP-l(5.626 cos cot -1.766ka sin oat) + 1.406P2_^ cos 2cot -0.883P-l2 ka sin 2u3t> (4.44) The suction force is obtained from equation (4.7), restricted to the first powers of ka, W2 = pc^a^ <1.554Po^ + 0.777pi2 + PoPi(3.107 cos cut + 1.888ka sin oot) + 0.111^2^ ^°s 2a)t + 0.944ka ?>-^ sin 2jiiA (4.45) The following expression is obtained for the total frontal resistance: MCA TM 1324 49 W = W2_ - W2 = pc2a2 ^1.25900^ + 0.630P3_2 + pQp.j_(2.519 cos act - 3.653ka sin (x>t) + O.SSOP^^ cos Sjot -1.827(3-^2 y^ ^^^ gcDtV (4.46) For the meaji value of the frontal resistance W = pc2a2 <1.259Po2 + 0.630P-l21 (4.47) The flapping wing is considered such that z = Pqx + p-]_ cos oot (4.48) In this case Zo(x^y) = - ^^0 Z(x,y) = - ikcp^ (4.49) Comparison of these expressions with equation (4.25) shows that in the case considered it is necessary to take cPn ikcPi (1 + ikGp) k^cp-, G^ A^ = ^ A = ^ ^ B = ^ ~ (4.50) Oi-Gq 1-G-L + ikGg 1 - G^ + ikGg that is, 1- ika 0.0468 ^ k^c pi(0.0947+0.156ika) Ao = l.lOScPo A = ikcPi 0.9053. o.202ika ^ = 0.9053 - 0.202ika (4.51) or, by restriction to small terms of the second order with respect to k, Aq = l.lOScPo A = ikcP-L(l.l05 + 0.195ika) B = 0.105k2cp-L (4.52) For the lift force P = pc2a^< 2.813Po + 2.813kP-L sin oit + 0.301k^aP-L cos ootV (4.53) eind for the moment of the pressure forces about the y-axls My = - pc2a3 il.473Po + 1.473kp-L sin cot -0.181k2aP3^ cos octl (4.54) 50 NACA TM 1324 The component of the frontal resistance {= Wj_ = Ppo = pc^a^ < 2.813Po^ + 2.813kPo3l sin <^^ + 0.301k^aPoPl cos (jotV (4.55) The suction force will be, with an accuracy up to terms of the second order with respect to ka: Wg = pa^c^ <1.554Pq^ + O.YYTk^P-L^ -0.57&^Q^-^^^a cos ojt + 3.107kPQ(3-|_ sin (Dt -O.TTVk^p-L^ cos 2ait> (4.56) For the total fro'ntal resistance W = pa^c^ •<1.259PQ2a -0.777k2p-^_2 -0.294kPQP2_ sin ODt + r2_Q Q n^r~ /,^+ J_ Pi 7'771,2q 2 0.677k^aPoPi cos cut + 0.777k'^P3_'^ cos 2oDt> (4.57) } Its mean value will be ,2_2 7t o<^qp. 2 _r\ 777i,2q_ 2 W = pa^c-^^ <1.259Pq'^ -0. 777k'^p3_'^y (4.58) so that a decrease is obtained in the frontal resistance as compared with the wing which does not execute a flapping motion. REFERENCES 1. Kochin, N. E. : Theory of a Wing of Circular Plan Form. Prikladnaya Matematika i Mekhanika, vol. IV, no. 1, 1940, pp. 3-32. 2. Schade, Th. : Theorie der schwingenden kreisformigen Tragflache auf potentialtheoretischer Grundlage. I Analytischer Teil. Luftfahrtforschimg, bd. 17, Ifg. Il/l2, 1940, pp. 387-400. WACA TM 1324 51 THEORY OF WING OF CIRCULAR PLAN FORM"' By N. E. Kochin A theory is developed for a wing of circular plan form. The dis- tribution of the bound vortices along the surface of the wing is con- sidered in this theory, which has already been applied in a number of papers. In particular, the case of the circular wing has been examined by Kinner in reference 1. A second method is considered herein which permits obtaining an expression in closed form for the general solution of this problem. The wing is assumed infinitely thin and slightly cambered and the problem is lifiearized in the usual manner. Comparison of the results of the proposed theory with the results of the usual theory of a wing of finite span shows large divergences, which indicate the inadequacy of the usual theory of the case under consideration. For the wings generally employed in practice, which have a considerably greater aspect ratio, a more favorable relation should be obtained between the results of the usual and the more accurate theory. 1. Statement of the Problem The forward rectilinear motion of a circular wing with constant velocity c is considered. A right-hand system of rectangular coordinates Oxyz is used and the x-ajcis is taken in the direction of motion of the wing. The wing is assumed thin with a slight camber and has as its projection on the xy-plane a circle ABCD of radius a with center at the origin of the coordinates (fig. 2, in which a section of the wing in the xz-plane is also shown). Let Ux,y) (1.1) represent the equation of the surface of the wing, where the ratio ^/a as well as the derivatives S^/dx and S^/Sy are assumed to be small magnitudes . *"Teoriya kryla konechnogo razmakha krugovoi formy v plane." Prikladnaya Matematika i Mekhanika, Vol. IV. No. 1, 1940, pp. 3-32. 52 NACA TM 1324 The coordinate axes are assiimed to be immovably attached to the wing. The fluid is considered incompressible and the motion nonvortical, steady, and with no acting external forces. The velocity potential of the absolute motion of the fluid will be denoted by a2 ; x < (1.7) z=+0 ^<^z^z=-0 ' ' which expresses the continuity of hp/bz in passing through the surface of discontinuity 2 . 54 MCA TM 1324 The dynamical condition expressing the continuity of the pressure in passing through the surface of discontinuity 2 is considered. In order to determine the pressure p, the formula of Bernoulli is applied to the steady flow about a wing obtained by superposing on the flow considered, a uniform flow with velocity c in the direction of the negative x-axis. In this steady flow the velocity projections are determined by the equations Sep S

a^ ; x < (l.lO) z=+0 ^^4=-0 which expresses the continuity of S(p/dx in passing through Z . The function ip suffers a discontinuity on the surfaces S and 2 , which means that along the surfaces S and 2 > surface vortices are located as shown in figure 2. The direction of such a surface vortex is perpendicular to the direction of the relative velocity vector of two particles of the fluid adjacent to the surface of dis- continuity on its two sides. In particular, on the surface 2^ °'^ account of equation (l.lO), only htp/by suffers a discontinuity and therefore the vortex lines on 2 ^.re directed parallel to the x-axis as shown in figure 2. NACA TM 1324 55 Since all the vortices lie in the xy-plane, at two points symmetri- cal with respect to the xy-plane, the values of h(p/hz will "be the same, whereas the values of B

a2 ; x < (l.l2) Finally, since the fluid far ahead of the wing is assumed to be undisturbed, the condition at infinity is . . Sep 9

0, which on the circle S satisfies the condition - c g (1.14) z=0 °^ (^l 56 NACA TM 1324 on the strip 2 , the condition (sL = ° t^-^^' 'z=0 on the entire remaining part of the xy-plane, the condition (p(x,y,0) = (1.16) and the derivatives of which remain bounded in the neighborhood of the rear semicircumference BCD, while in the neighborhood of the forward semicircumference BAD they may approach infinity as l/v^ where 5 is the distance of a point to the semicircumference BAD. Finally the condition at infinity (l.l3) must be satisfied. An expression for the harmonic function (p(x,y,z) is given in closed form depending on an arbitrary function f (x,y) satisfying all the imposed requirements besides equation (l.l4). The function ^(x,y) can be determined from this condition, that is, the shape of the wing corresponding to the function f(x,y). An integral equation is also given, the solution of which is reduced to the determination of the function f(x,y) for the given shape of the wing, that is, for a given function ^(x,y). 2. Derivation of the Fundamental Equation Inside the circle ABCD, the point Q with coordinates ?, t] is taken and the function K(x,y,z, ^,tj) constructed, where x,y,z are the coordinates of the point P, according to the following conditions: (1) The function K, considered as a function of the point P, is a harmonic function outside the circle ABCD. (2) The function K becomes zero at the points of the plane xy lying outside the circle ABCD. (3) The derivative Sk/Bz becomes zero at all points of the circle ABCD, except the point Q. (4) When the point P approaches the point Q, remaining in the upper half- space z > 0, the function K increases to infinity but the difference K - (l/r), where r = ^(x - ?)2 + (y - Ti)2 + z^ remains boiinded. NACA TM 1324 57 (5) The function K remains finite and continuous in the neigh- borhood of the contour C of the circle ABCL. Because of the second condition, the values of the function K at two points situated symmetrically with respect to the plane xy differ only in sign: K(x,y,-z,?,Ti) = - K(x,y,z,4,Tl) (2.1) as follows from the principle of analytic continuation. It is. then evident that if the third condition is satisfied on the upper side of the circle ABCD it will he satisfied also on the lower side, since according to equation (2.1) the derivative 5k/Sz has the same value at two points situated symmetrically with respect to the xy- plane. It is evident further that when the point P approaches the point Q from below so that z< then K{x,y ,z,^,t\) will behave as - l/r. Because of the third condition, the function K can be continued into the lower half-space through the upper side of the circle ABCD as an even function of z. Thus a second branch of the function K is assumed, again determined over all the space outside the circle ABCD and differing only in sign from the initial branch of the function K. It is then evident, however, that at the points of the upper side, of the circle ABCD, the values of the second branch of the f\anction K and its derivatives coincide ■'/rith. the values of the first branch of the function K and its derivatives at the points of the lower side of the circle ABCD. That is, in the analytic continuation of the second branch of the function K through the upper side of the circle ABCD into the lower half-space, the initial branch of this function is again obtained. A two-sheet Riemann space is considered for which the branching line is the circumference ABCD. In this space K(x,y,z,C,T)) is a single-valued harmonic function remaining finite everywhere with the exception of the two points Q having the same coordinates (C>T>0)> but belonging to two different sheets of space; in one sheet the function K behaves near the point Q as l/r and in the other sheet as - l/r. Such a fionction K(x,y,z,Cj'n) can readily be constructed by the method of Sommerfeld (reference 2) . In this way for the case of a two-sheet Riemann space having as branch line the z-axis, a harmonic function v(p,= — /ya-T 'ya+Tl Jtr V = — = — arc tan — ^==^ jtri A/a-T A/a+T| Jtr _ycr2 - t2 or finally 2 ^Vp^ "'" 2 V = — arc tan rtr r An inversion with respect to the point with coordinates p = a,

0, where r=V(x - O^ + (y - Ti)2 H- (2.5) a/7'2 2 2 2x2 ,.22 a//'o2 _^ ^2 _, 2~~ 272 . 2/ 2 , 2v = Y(a -X -y -z) +4az = 'y(a +x +y +z) -4a(x +y) R That this function satisfies all the above set requirements is easily verified; the arc tangents must be taken between and Tt/2; for z < the value of the function K is obtained by equation (2.1), The following function is set up: (P^(x,y,z) = ^ J J* K(x,y,z,?,Ti)f(?,Ti)dgdTi (2.4) where f(x,y) is an arbitrary function, which is continuous together with its partial derivatives of the first and second order in the entire circle S, and the integration extends over the entire area of the circle S. Evidently, (p-,(x,y,z) is a harmonic function in the entire space outside the circle S. Because of the first property of the function K, the function (p-,(x,y,z) becomes zero at all points of the plane xy which are outside the circle S. Hence equations (1.15) and (l.lS), which must be satisfied by the solution \/a^ - g^ - r^ Va^ - x^ - y^ - z^ + R fx - g x1 '^ 2a2r2 + (a^ - g^ _ T)2)(a2 - x^ - y2 _ z^ + r) (^ r^ Rj '2.5; If a point with coordinates x^y^z is near the contour C of the circle S the distance of this point to the contour C is denoted by 6j then & =Va2 + x2 + y2 + z2 - 2a Vx^ + y2 (2.6) Hence near the contour C, the approximate equation holds: R =. 2a& (2.7) When the fixed point g,T] lies inside the circle S while the point with coordinates x,y, z lies near the contour C of the circle, then, as follows from equation (2.5), ^ = _ rv^Va^ -l^ - ^ Va^ . x^ - y^ - z^ + r + o(l) (2.8) ox nar'^R where the symbol 0(l) denotes a magnitude which remains finite when 5 approaches zero. Thus Sk/Sx has the order l/-y/&. The principal part of Sk/Sx is not a harmonic function. It is not difficult, however, to find a harmonic fionction having the same infinite part near the contour C as Sk/Sx. For this, it is sufficient to form, after the analogy of equation (2.5), the derivative Sk/SC; this derivative remains finite near the contoiir C of the wing; moreover it is easy to see that Sk ^ _ 2V2 aVa^ - x^ - y^ - z^ + R ax aC ^ [2a2r2 + (a2 - ^2 _ ■^){^^ _ ^2 _ y2 _ ^2 + r) ] NACA TM 1324 61 This function is harmonic and differs from Sk/5x by a quantity which remains finite near the contour C. By computation, it is further shown that the function just described is represented in the form of the integral 3k hK 1 n Va^ - «^ - 1^ Va " ^ " 7 - z^ + R cos r dr ^ + "51= - ~T~1= J — ; — ; — ^-^ ' ; — ; — ; (z.io) " V^ -n (x'^ + y'^+ z'^+a'^- 2ax COS r- Say sin r)(5 +n +a'^- ZaS COS r- 2an sin r) where the function ^/ a^ - x^ - y^ - z'^ + R "P 9 9 9 X + y + z + a - 2ax cos y - 2ay sin y is a solution of the equation of Laplace having the circumference C as the branching line and the point with coordinates (a cos y, a sin y, O) as a singular point. From this it follows that the function 3h hK 1 y Va^ - K^ - -r?- Va^ - x^ - y^ - z^ + R cos r dr ^ n^V^/ (x^ + y^ + z^ + a^ - 2ax cos x - 2ay sin r)(5^ + t]^ + a^ - 2ag cos r - 2a7) sin r) 2 n ^ 1 P Va^ - K^ - .n^Va^ - x^ - y^ - z^ + R cos r dr 3£ iT^pZ I (x' + y^ + z^ + a^ - 2ax cos if - 2ay sin r)(6^ + t)2 + a^ - 2a5 cos r - 2aii sin y) "2 remains finite near the points of the rear semicircumference of the circle S. Therefore it is assumed hf 1 / / , ,/aK 3>t S 1 2 Va^ - ^ - n^ Va*^ - x'^ - y^ - z^ + R cos y dy n^-jzj (x^ + y^ + z^ + a^ - 2ax cos y - 2ay sin r){K^ + t)^ + a^ - 2a5 cos y - 2aT) sin y)l ►dSdn (2.12) 3ii 62 NACA TM 1324 Integrating with respect to x and considering the condition at infinity (1.13) yield the final equation ^(x.y.z) = 2^ J J f(C,T)) i a^ - x^ - y^ - z^ + R G(r) cos j n 2 2 2 2 o n - dx dr (2.17) X + y + z'' + a'' - 2ax cos y - 2ay sin r The given functions F{^,r\) in the circle S and the function G(r) in the interval (-«/2, n/2) completely determine fC^^Tj), so that the equations (2.13) and (2.17) are equivalent. The equation (p(x,y, z) obtained satisfies the conditions imposed in section 1. This function is evidently a harmonic function in the entire space exterior to the circle S and satisfies the conditions at infinity, equation (l.is).. From equation (2.12) it follows, that in the plane xy for x^ + y^ > a^ the condition is satisfied: (I) and from equation (2.13) it follows that = z=0 (x,y,z) at the points of the rear semi circumference C and to determine the behavior of these derivatives on approaching the points of the forward semi circumference C. In considering the neighborhood of the rear side of the circum- ference C, equation (2.16) may be used. The latter shows that Scp/Sx remains continuous at the points of the rear half of the circumference C and becomes zero at these points. The behavior of the derivatives with respect to y and z of the following function is considered: 64 NACA TM 1324 #(x,y,z) = r rK(x,y,z,4,Ti)F(^,Ti)d4dTi (2.18) near the contour C. = // If F(?,n)d? dn Similarly to equation (2.9), + 2aV2 V^^ - x^ - y2 - z^ + R (2.19) Sy St] n[2a2r2 + (a^ - ^2 _ Ti2)(a2 _ x^ - y2 _ z^ + R)] (2.20) •2 T,2 yVa^ - g'^ - T]^ T] ya^ _ ^2 - Ti'-^ R and similarly to equation (2.14), //|F(?,T,)dgd,= - //K^dgdr, (2.21) where this part of the integral remains finite everywhere and on the conto\ir C becomes zero. In order to evaluate the remaining part of the integral equa- tion (2.19), the following two integrals are considered: Ji(x,y,z) "\ Va^ - g^ - Ti^ dg dTi 2a2r2 + (a^ - g^ _ T)2)(a2 - x^ - y2 - z^ + r) J2(x,y,z) = ^ (2.22) dg dTi Va^ - g^ - n^ ^a^r^ + (a^ - ^2 _ T]2)(a2 . x2 - y2 . z2 + r)J y Vx^ + Both, on account of the symmetry, depend only on 'Vx'' + y^ and z; hence without restricting the generality, it may be assumed that y = 0, X > 0. The distance 5 of a point with coordinates (x,0,z) is introduced to the contour C: MCA TM 1324 « 65 S =Y(a - x)2 + z2 Since R 5- Ix^ + z^ - a^ the following relation will hold: 2a2[(x - 5)2 + TjS + z2j S Polar coordinates are introduced C = p cos ^ ; r\ = p sin i^ whence ,a ^2 It Jl(x,0,z) < ' ' pJ/^2T72^d^ Since 2rt r 2a2 jp2 _ 2px cos ^ + x2 + z2j (ii& 2rt p2 - 2px cos ,S + x^ + z2 y(p2 + ^2 ^ ^2^2 _ ^^2 ^2 hence J-|_(x,0,z)^ i^ 3. pV^ - P^ d-p A/(p2 + x2 + z2)2 _ 4p2 x2 For x ^ a Jl(x,0,z)< JL r Pya2-p2dp ^ ^ ' V(p2 + x2)2 _ 4p2 x2 a'- ''0 _ jt_ I p Va2 - p2 dp < _rt_ / pdp rt 66 NACA TM 1324 While for x -^ a, use is made of the inequality R > a^ - x2 - z2 to obtain Jl(x,0,z) < i ^jlJ~~^~~^ d^dTi 2 '-'g^ a2[(x - 5)2 + 1)2 + z2] + (a2 - ^2 _ T]2)(a2 - x2 - z2) -2 It 1 / / p Va^ - p^ d^ dp — I I ^ (2.23) 2 / / a* - 2a2 xp cos ,J + p2(x2 + z2) ^0 ^0 a p*\ja^ - p2 dp / p ya^ - p2 dp / pdp y[a4 + p2(x2^z2)]2_4a4x2p2 " J^ a* - x2p2 ^ J^ ^g-^/T /Q wi "• ' M \-^ ' ^ / 1 " ^'-^ -^ H wfi un a ya - p The following inequality results : jt Jl(x,y,z) <- (2.24) The second integral is considered. As before, a J2(x,0,z)<^ ''' ^^ a2 Va2 - p2V(p2 + x2 + z2)2 . 4p2x2 For X ^ a For x-< a an inequality of the type in equation (2.23) is used: P dp J2(x,0,z).$ It 'o VCa^ - p2)[a^ + p2(x2 + z2) + 2a2xpjLa2 - xp)2 + p2z2j ^ JL I p dp ^^Jo '\/(a2 - p2)[(a2 - xp)2 + z2p2] MCA TM 1324 67 If z ^ a - X and therefore 6 < z Vs , then J2(x,0,z) dp a'^z rt2 ^2 ;0 Va2 - p2 2a^z ^25^2 but if < z ^ a - X, and therefore 6 •$ (a - x)-v/2, then p dp ^ ^ I P d-p J2(x,0,z)^^/ " -^^ <-^ / " 7 =-^ ^^ ^1 (a2-xp)Va2-p2 a^J (a - x) Va2 - p2 a2(a-x) a26 '0 The following approximation is obtained: J2(x,y,z) ^6V2 vhere S ='Y(a - Vx2 + y2)2 + z2 Near the contour C (2.25) (2.26) R ' 2a& If this relation, the evident inequality (2.27) |a2 - x2 - y2 - z2 I « R and the obtained approximations are used, the following approximation is obtained from equation (2.20): x/d^^)^'-'^^^^ -(^) It is evident from equations (2.19) and (2.2l) that near the contour C (2.28) 68 NACA TM 1324 The following derivative is formed: II = XX|^F(?'^)^?^T But Sk ^ 2z_ where , .2 A arc tan Ah If 1 + A^ z z(a^ + x^ + y^ + z^ - R) ^^ rR(a2 - x^ - y2 _ z^ + r) A = ya^ - g^ - 1)^7 a^ - x^ - y^ - z^ + R ar-y^ Hence if |F(g,n) I < M then, on account of the inequality A the approximation results; b^ « 2M r r ^ dK dT] + 2,-^ azM(a2 + x^ + y^ + z^ - R) jtR Va^ - x^ - y^ - z^ + R " g 4 a^ _ ^2 _ ^2 d g dr| 2a2r2 + (a^- g2 _ T)2)(a2 _ x^ - y2 - z'^+ R) Noting that S and making use of approximation (2.24) yield NACA TM 1324 69 S* < 4jtM + 2 V2 M z(a^ + x^ + y^ + z^ - R) rVs^ - x^ - y2 _ 72 + p Since for z > z _ z VR - (a^ - x^ - y^ - z^ ) ^ ^ 4rtM + ^^^ (a2 + x2 + y2 + z2 - R) Vr - a2 + x2 + y2 + z2 aR Now when the point P(x,y,z) is near the contoiir C> then because of R = 2a6 ; I x2 + y2 + z2 - a2 I ^ R there is obtained a-i- '& (2.29) Equation (2,16) is again considered. Since the derivatives _L'\^2 _ ^2 _ ,,2 _ ,2 ra-^ - x^ - y- - z' yVa2 - x2 - y2 - z2 + H ^ + K = - ■ < R Va2 - x2 - y2 _ z2 ■~~r z(a2 + x2 + y2 + z2 - r) + n = — ' — - RVa2 - x2 - y2 - z2 + R )(2.30) = (a2 + x2 + y2 + z2 - R)VR - a2 + x2 + y2 + z^ 2aR ^ ' ^ -/ have near the contour C the order l/-\Jb, it is clear from equa- tion (2. 16) and the obtained equations (2.28) and (2.29) that at the points of the rear semi circumference of C there is the estimate S^

while the latter two of these estimates are obtained in a similar manner from equation (2.13). In this manner all the conditions which must be satisfied by the function (p(x,y,z) are satisfied. The shape of wing to which the obtained solution corresponds is explained. By equation (l.l4) Hence it is necessary to find the value Scp/Sz in the plane of the circle S. Both sides of equation (2.13) are differentiated with respect to z and then z set =0. On account of the very definition of the function K, NACA TM 1324 71 ^K ' ' ^ lim / / ^ f(C,Ti)d5dTi = lim / / "ST" ^^^^'l)^? ^^ = - 2«f(x,y) S S (2.35) Moreover, on account of equation (2.30), (^ for x^ + y^ < a^ ;^ A-;^ ^ n I lim . dz • " \ .Vx^ + y2 - a^ z-*+0 ^ I I - - for x2 + y2 > a^ If this is taken into account, (^) = - f(x,y) + g(y) (2.36) where 8{y) Va^ - g^ - ri^ cos rf(£>Ti)dr dx d£ dii y^ - a^(x^ + y^ + a^ - 2ax cos r - 2ay sin r) C? + ti^ + a^ - 2a5 cos r - 2aTi sin r) 2 (2.37) For the fimction ^(x,y) the following expression is found: X ^(^^y) = 7 vT f(x.y)^ - ^ ^ + si(y) (2.38) c u^ where gi(y) is an arbitrary function of y. Thus, for the assumed degree of approximation, the bending of the wing in the transverse direction produces no effect on the form of the flow. It is assumed that the shape of the wing is given, that is, the function ^(x,y) and therefore the following function are given: c ^ = M(x,y) (2.39) 3x 72 NACA TM 1324 From equations (2.34) and (2.36) it is clear that f(x,y) = M(x,y) + g(y) (2.40) Substituting this value in equation (2.37) and introducing the notations N(y) = -^ „Va^-y' Va^ - g2 - 1)2 M(g,Ti) cos y dr dx dg dn O u J^„ d V^^ + y^ - a^(x^ + y2 + a^ - 2ax cos r - 2ay sin r)C5^ + ri2 + a^ - 2aC cos r - 2aT) sin r) ^ (2.41) H(y,Ti) = -^ " ^^ J^^ T Zn3 y ^2 Ya2 - ^2 - T)2 cos y dy dx d^ J~2 2^+«' On V^^ + y2 - a2(x2 + y2 + a2 - Sax cos y - 2ay sin r)(£^ + 1)2 + a2 - 2a5 cos r - 2ail sin y) give an integral Fredholm equation of the second kind for the determi- nation of the function g(y) : ^f g(y) = N(y) + H(y,Ti)g(T])dTi (2.42) In Consideration of examples^ a function f(x,y) shall be given and the shape of the wing then determined by equation (2.38). For the obtained shapes of the wing it is not difficult to find a solution by the usual theory, a fact which provides the possibility of evaluating the degree of accuracy of the usual theory. 3. Computation of the Forces Acting on the Wing The fundamental equation determining the motion of the type \ander consideration is recalled: '(=''^'^>=^ rj'f^'''=''^'^''''^7vi'' ri ^ ^ ^ (3a) yB.^ - ^ - vi^'J&^ - x2 . y2 . z2 + p cos r dr dx —— ■ '■f(?,'l)d,^dil (x2 + y2 + z2 + a2 - 2ax cos r - 2ay sin r)fe2 + t)2 + a2 - Z&E, cos r - 2ari sin 2 MCA TM 1324 73 The value of the function

a^ , x < (3.2) Z-H-0 Then evidently lim ip(x,y,z) = - a^ , x< (3.3) 2r*-0 The circulation over the contour M'NM (fig. 2) connecting the two points M and M' of which point M' lies on the lower and point M the upper side of the half-strip Z, both points M and M' having the same coordinates x,y,0, is denoted by r(y). It is then evident that r(y) =$(M) - a sin y. The following expression is written for the distribution of the circulation in the vortex layer formed behind the wing: r(y) = - 4 iff V-' -'"■'' '^'''^ f MH^IL . A cos r dr a, a, (3.7) " J J J (5^ + l2 + a2 - 2a5 COS r - 2an sin r) U I ^ ^^"^ '^ " ^ I J S 2 The forces acting on the wing are computed. Denoting by p_|_ the presstire at a point of the wing S on the upper side of the wing and by p_ the pressure at the same point on the lower side gives on the basis of eq\iation (I.8) P_ - P+ = - 2pc ^ (3.8) 5x where the value of btp/hx is taken on the upper side of the wing. , For the lift force P, the following expression is obtained: rVa2-y2 I -2 djc dy Va^y 2 a a - 2pc I [^(Va2 - y2,y,0) - (p(-Va2 - y2,y,0)]dy = 2pc / ^(y) dy The following formula is obtained: a r. p = PC J r(y)dy (3.9) -a NACA TM 1324 75 having the same form as in the usual theory of a wing of finite span. But the distribution of the circulation r(y) by the present theory is somewhat different from that obtained by the usual theory. The derviva- tion given is not connected with the shape of the wing. With the aid of equation (3.6) P may be directly expressed through 3it p=.^ Le.2 -?2 - n2 f(C,ri) / ; , °°' ^ ^^ -d^dn (3.10) ^2 J J / g ^ + T)^ + a'^ - 2a^ cos y - Sar) sin r S '^ 1 2 The expression for the induced resistance W in terms of the circulation r(y) likewise has the same form as in the usual theory: ^-l f f r(y) M^ -i-, dy dy. (3.11) because the origin of the induced resistance is due to the fact that behind the wing a region of disturbed motion of the fluid is formed; the kinetic energy of this disturbance is determined on the other hand exclusively by the distribution of the circulation at distant points from the wing. The expression for the induced resistance is obtained from the momentijm law. A surface enclosing the wing S is denoted by B; the momentum law applied to the wing in a steady flow then leads to the expression W = j j P cos(n,x)dcT + \ I pVn'^x^'^ (3.12) B B where n is the direction of the outer normal to the surface B and V^, Vy, V^ are the components of the velocity in the relative motion of the fluid about the wing. Thus VY=-c+t--; V„ = -c cos(n,x) + — - ^ dx " bn hm p IT ^ m ^ (I ■T 76 NACA TM 1324 Substituting these values in the preceding formula and noting that r rcos(n,x)da =0^ rr^da = B B results in B ■- -• B The surface B consists of a hemisphere of large radius with center at the point x = Xq < -a of the x-axis enclosing the wing, and of the circle cut out by this hemisphere on the plane x = x^. With increase in the radius of the hemisphere to infinity the corresponding parts of the integrals entering the preceding formula approach zero. On the surface x = x, f \ -, . o(p Sm cos(n,x) = - 1 , — r= _ ^2- on ox therefore 'rM[0^(^:f-(gj' W = i:^ i .1 f^l + (^1 - (^1 dy dz where the integration extends over the entire plane x = x ^0 the following equation is obtained : v2 />^v2 0" w = f rr f ) ^ (n )> dz (3.14) For (3.15) where equation (3.11) is obtained. J_ r cir(y.') 2« ^.a y' - y (3.17) NACA TM 1324 77 In order to find the center of pressure, the principal moments of the pressure forces about the Ox and Oy axes are determined. For the moment about the Ox axis, Mx a = J J (P_ - P+)y dx dy = - 2pc J J — y dx dy = 2pc J (y)y dy S S ""^ from which Mx = pc J yr(y)dy (3.18) -a Expressing M^ in terms of f(x,y) yields 3rt 4 pca^ I P P Va^ - g2 _ ^2 f(g^T])sln r cos r dy dg dr^ (g^^g) 2 n / / / g2 + 1^2 + a,2 - 2ag cos y - 2a.r\ sin y For the moment about the Oy axis, - ^' s s Substituting the value S

is determined hy the equation dp r 2np dp ^ 2rta (4.13) + p2 - 2ap cos(^ - y) J Ya2 - p2 Substituting this value in equation (4.3) yields 3rt r (y) = - 2aca I cos j 2 a(i ^ si" r) - 1 I a sin r - y I d-c If the integral is taken, r (y) = — V 4a + 2 V2a(a - y) + 2 V2a(a + y) - , , . ^ V2a - Va - y (a + y) log -»— '^ V2a + Va - y , , A/2a - A/a + y 1 , - (a - y) log -5L_ W y (4.14) V2a + Va + y J Setting y = -a cos 9 and expanding r(-a cos G) in a trigono- metric sine series in the interval < < n give after simple computations r(-a cos 0) = ^- 4 + 4 cos — + 4 sin — 9 2 9 2 1 - cos _ (1 - COS 9) log - (l + cos 9) lof 1 + cos £ 2 1 - sin I 1 + sin — = A sin 9 + A^ sin 39 + A^- sin 59 + (0 < 9 < n) (4.15) NACA TM 1324 81 where 16aca 4aca /l 1 1 \ — ^ ' ^sk+i = - ■^— ( - + - + • • • + - — - ; Jt^ n2k(k + 1 (2k + 1 \3 5 4k +1/ H (k + l)(2k + 1) (k = 1, 2, ..) (4.16) so that 16aca . 496aca ^3 = - -^ i ^5 = ■) 45it2 4725jt2 The distribution of the circulation obtained is very near that of an elliptical distribution. The lift force and the induced drag are obtained by application of equations (4.10). P = — pcaAi = — pa^c^a « 2.5465 pa^c^a 2 jt (4.17) W = - 7tp(A-L2 + 3A32 + ...) a 1.034 pa2c2a2 In order to determine the position of the center of pressure, M^ must be computed by equation (4.8). Equation (4.13) gives My = - 2 pc'^a-^a ; X(, = - -^ = - a (4.18) The distance from the center of pressure, which evidently lies on the Ox axis, to the leading edge of the wing thus constitutes about 0.238 of the diameter of the wing. In order to determine the shape of the wing corresponding to the assumed function, it is necessary to form the function g(y) by equa- tion (4.12). If equation (4.13) is considered. 3n 2 Jp^ r\2 ^ ^ , . a ca P I cos r df dx g(y) = — 2" n'- jj Vx2 + y2 _ a2(x2 + y2 + a2 - 2ax cos j - 2ay sin y) (4.19) 82 NACA TM 1324 The computation shows that for x > ^a - y 3jt cos X dy _ _ ffx rt x^ + y2 + a^ - 2ax cos y - 2ay sin y 2a(x2 + y2) 2 V ^ x^ + (y - a)2 log ^^i '— + 2a(x2 + y2) x2 + (y + a)2 x(a2 + x2 + y2) ^^^ tan ^^ + y^ ' ^^ (4.20) a(x2 + y^)(x2 + y^ - a^) ^ax If y = - a cos 6 ; Va^ - y^ = a sin d Ho(9) = — sin 9g(-a cos 9) ^ ac (4.21) for < 9 < rt "jsin f 2 4. _ sin e ) nt cos 9 ^ +^ ^°^ ? ^ ~ '^. Vt2-sin2e 1 2(t2+cos2e) " 2(t2+cos20) ^°^ ^\^ ^^^A 9 "*■ t(t2 + 1 + cos29) t2 _ sin2t ^ • arc tan > dt (t2 + cos20)(t2 - sin29) 2t Computation of this integral gives \2 / 9\2 1 + sin -S \ 1 / 1 + cos - — sin 91 log ^ I + - sin 9 ( lo^ * 1 - sin _/ \ 1 - cos —J 2^^ \ 2 - 1 - sm — 1 - cos — cos I log 1 + sin I log 1 (4.22) '^ 1 + sin _ "^ 1 + cos - 2 2 MCA TM 1324 83 The shape of the wing is thus determined by the equation tan y ^(x,y) = ox dy - 1 / . _ __ -JzE + Va + y V _ _J_ (-1 ^„ V2a + ^a - y 8rt^ log V^a - v^ + yy Bn^ V2a - V^^^ V2a log V^ - V^ +J^ dZH log V^ - V^ - .y > (4.23) 2jt2-^a + y -yi^ + V^ + y 2.tC-^Jb. - y -^/Ha + ^a - y This wing differs little from a plane wing inclined to the xy-plane by a small angle a and may be obtained from such a plane wing by twisting. The values of the function ^(x,y) for the mean value y = and for the values y = i a/2 are ^(x,0) = ax; 1 2 2 ^- arc tan y Jt" / Vl - y2 ^ dy - -^ log2(V2 + 1) + 2V2 log(V2 + 1) ^-^t) ax 1 2 — ^- —7^ 0.8452 ax 1 . -2/0 , r?\ 1_ ._„2 arc tan y ± ^ i;,„ , r=-. Yi _ y2 2^2 log^3 + 8n' ^^Vs log(2 + VS) + ^log 3 0.8335 ox It is of interest to consider what results for the obtained wing are given by the usual theory. The circulation obtained by this theory is denoted by rQ(y); if the expansion of this circulation in a trigono- metric series is TgC-a cos 0) = B3_ sin 9 + Bg sin 20 + . , . (O < < n) (4.24) then the usual theory gives an equation for determining the coefficients B^j which in the case considered reduces to the form 84 NACA TM 1324 l_j Bn sin n0 = 2«ca sin 9 3^ = - 0, (3^ = - 0.0460 i pg = - 0.0212, 1213 Equation (4.26) shows that 4aca(jr2 - P]^) Bl = ;; ; B2k = J B2k+i 4acaPg^^^ 2 + j:(2k + 1) (k = 1, 2, ...) (4.28) The numerical values of the first coefficients will be B^ = 2.4784 oca ; B3 = 0.0562 oca ; B5 = 0.0087 aca By = 0.0024 aca ; Bg = 0.0009 oca, ... MCA TM 1324 85 The following value is obtained for the lift force : Pq = - npcaB-]^ = 3.8932 pc^a^a (4.29) exceeding the accurate value by 53 percent. For the induced drag. Wq = 2.416 pa2c2a2 (4.30) with an error of 134 percent. 2. If a is assumed to be small, f(x,y) = - 2c(xx is taken. The circulation r(y) is computed. First the value of the fol- lowing integral is found. ■'{^2 .2 „2 gVa - £^ - Ti^ dr dTi 4 o f.^^s ^-i 2 ! S ! = Z na^cos x (4. 31) ^2 + 1^2 + a^ - 2a^ cos x - 2aT) sin y 3 Equation (4.3) gives 3jt »2 r(y) = -:: — / cos^r 3rt Jt 2 ' a(l ± sin x) |a sin r - y| dr The computation of this integral leads to the very simple expression r(y) = 2ca(a2 - y2) (4.32) Thus in the case considered, a parabolic distribution of the circu- lation was obtained. For this reason the computation of the forces can be easily carried out: P = a, pc I r(y)dy = — 0Lpc2a3 « 2.667 cxpc^a^ -a (4.33) W = - pc^a^a^ » 1.2732 pc^a^a^ It 86 NACA TM 1324 Equation (4.3l) is used in the computation of My by equation (4.8): Mv = — pc^a^a- 1.509 pc^a^a ; Xp=--^=- — a (4,34) y 27« P 9n In order to determine the shape of the wing it is necessary to compute the function g(y)j equation (4.12) yields g(y) = 4(xca" 17" I — ^ -(Va2-y2 p2 cos IT df dx 2 Vx^ + y2 - a^ (x2 + y2 + a^ - 2ax cos y - 2ay sin y ) Setting 3n Ei(e) = sin eg(-a cos 0) (o < ^ it) 4aca (4.35) and carrying out the integration with respect to y yield ^sin 9 sin e y . /,.2 , 2, Hi(e) = (t2 + cos20)2 Y^2 _ sin^e t(t + cos'^e) + I t2+ 4 cos^^ -= - (cos^e - t2)(t2+ 1+ cos20) - - t cos ©(t^ + 1+ cos20)io t'^+ 4: sin' 4 2(t2 4 cos20)2 + ^^2 _ cos20 ) [ i + (t^+cos^ 0) 2 J 2(t2 - sin20) Integration yields H-,(0) = — ^sin 0(1- sin - - cos arc tan- t2 - sin^ 2t •dt (^ L..„(ilI!li)_(iJ_!!ll) (1 - cos I) (1 - Sin I) - 1 + — log 4- 2/ 12 ^ r sin cos log tan— + - log 2 4 ^ (1 H. cos I) (1 - sin !) • (1 - cos IJ (1 + sin I) ■(4.36) In equation (4.11), the following is taken: gl(y) = a(a^ - y^) 2 ,.2' NACA TM 1324 87 Then for the function ^(x,y), which determines the shape of the wing, the following expression is obtained: ^(x,y) = a(a2 - x^ - y^) + 2aax ] W a + y / a - y _1_ (-/2a + ^a - y)(-/2a + -^a. + y) n I " V 2a "V 2a ^ 12 °^ (^2^ _ Virr7)(V2^ - -^fTT^) ~ 1 logA/m - 2- log (V2^ + VirriF)(Y2i - v^i^l ^ This wing is thus obtained as a deformation of the wing: C(x,y) = a(a2 - x^ - y2) which for small a differs little from a segment of a sphere. In particular, for y = 0, ^(x,0) = a(a2-x2) + i^ jl - ^ + ^ log(V2 + l)]» a(a2 - x2 - 0.0767ax) In order to apply the general theory to the obtained wing H-, (0) is expanded into a trigonometric series: Hi(0) = I '^ - ^ - I log tan I dx I sin + OP y^ sin(2k + 1)9 k=l 4k(k + l)(2k - l)(2k + 3) - 12jrk(k + 1) + 2(l6k2 + 16k - 3) I 1 - i + - - ... + — ) + 6(2k + l) ^ ' \^ 3 5 4k + 1/ ^ ' =£ r2k+isin(2k + l)e (4.38) k?=0 where r-L = - 0.6931 ; Y-3 = - 0.1783 • y^ = - 0.0812 Tj = - 0.0463 ; Tq = - 0.0300, ... QQ NACA TM 1324 For the case considered, the usual theory gives for the determination of the circulation r Q(-a cos 0) = B-,^ sin + Bg sin 29 + . . . (O < 8 < Jt) the equation > T, . no ■ n J -a sis- COS 9) 1 ) ^ sin nd\ / , B„ sm n0 = 2jtca sm < oa sm - -^^ - / / nBj^ / ^ 1^ c 4ca n=l sin 0j (4.39) Equation (4.35) and oo sin20=-8>J ^^^(^^ + ^)Q (O<0 f Soa^c IGoa^c 1 . /o, .\r. ^_^ Bn(l - -) = LAc-5^ ^2k+l - (2k- l)(2k-^ l)(2k + 3)J^^^(2^ ^ ^^' (4.41) from which without difficulty B^^ is obtained, in particular Bgk = ; B^ = 1.8457 aca^ ; B3 = - 0.2132 aca^ . B5 = - 0.0250 cica^ ; By = - 0.0075 (xca^ ; Bg = - 0.0032 cxca^, ... The following value is obtained for the lift force: P = i 3tpcaB3_ = 2.899 ac^a^p (4.42) exceeding the accurate value by 8.7 percent. The induced drag W = 1.3927 pa^c^a^ (4.43) exceeds the accurate value by 9.4 percent. NACA TM 1324 89 3. In order to give an example of a nonsymmetrical ving, f(x,y) = acy In this case it is first necessary to compute the integral n Ya^ - g^ - T]2 dg dT) g2 + r|2 + a2 - 2ag cos x - 2aT] sin ^ jta^sin r (4:. 44) On account of equation (4.3), 3jr ^2 r(y) = 4(xca^ 3n sin X cos X fn 2 l a(l ± sin r) I a sin r - y| dr After computing the integral, r(y)=^ (a + y)V2a(a - y) - (a - y)V2a(a + y) + i (a + y)(a - 3y) log V^ ' V^^^^ . 6 V2a + Va - y i (a-y)(a.3y) log V^ " V^"^ -y/2a + -y/a + y_ (4.45) is obtained. Assioming y = -a cos 9 and expanding in a trigonometric series give r(-a cos 0) = (xca' 9 9 2(l - cos 0)cos — - 2(1 + cos 9)sin — 9 1 _ cos — •| (1 - cos 9)(l + 3 cos 9) log 1 6 1 + cos — 2 1 - sin - 2: (1 + cos 0)(l - 3 cos 0) log 1 6 1 + sin ^ 2 J = Ap sin 20 + A4 sin 40 + . . . (O ^ -^ n) (4.46) 90 NACA TM 1324 where A2 = - 128 (xca^ 27jr2 (4.47) 4aca2 8k2 + 1 fl.i. 1 n2 6k(k2 - l)(4k2 - 1) V 3 (k = 2,3, ...) 4k - 1 1-V — ^^ 1 - 1/ (k2 - l)(4k2 - 1)J so that Ag = - 0.4803 aca2 ; A4 = 0.00549 aca2 Ag = 0.00234 aca2 ; Aq = 0.00123 aca2 Evidently there is no lift force, whereas for the induced drag the following value is obtained: W = 0.1813 pa2c2a^ (4.48) The moment of the forces about the Ox axis is: , Mx = - J rtpca2A2 = 0.3772 pac2a* (4.49) The moment of the forces about the Oy axis is computed with the aid of equation (4.8), where use must be made of the result (4.44), and it is found that My = (4.50) The following function is now computed: I ^ , . 2a ac / / sin r cos r dr dx g(y) = — ^ / / . ■ = ^^ I J^ Vx2+ y2 - a2(x2 + y2 +a2 - 2ax cos r - 2ay sin r) 2 Setting 32 Hp(e) = sin 0g(-a cos 9) (0< < it) (4-5l) '^ 2aac MCA TM 1324 91 and carrying out the integration vith respect to y give ^sin 9 H2(9) = / — :: ^^ ^ , ^ ■ ^ {- cos 0(t^ + cos^e) + C/OD (t2 + cos2e)2 Vt2 - sin20l ■^ t cos e(t^ + 1 + cos^e) + 2 ^ ' y (t^ + 1 + COs20)(cOs29 - t^) log t^ + 4 cos^ — t2 + 4 sin* £ 2 t cos 0L1 + ("t^ + cos20)2j ^ t^ - sin^© -arc tan t^ - sin'^ 2t ■dt (4.52) ) 5 9 , r- \ C a^rc tan y 3^ I = sin cos e<| log^(V2 + 1) + 3 J -y==| dy - -^ >■ , . , 1 + cos sin 0(1 - 5 cos 0) / 3^Qg 2 16 V 1 . COS 0\2 Sin 0(1 + 3 cos 0) I ^r,p 16 0v2 1 + sin — 1 - sin ^ 2 - 1 + 3 cos 1 + cos - ^ ^^^ 2 i°e ^ 1 - cos - T •z o a 1 + sin — 1-3COE0 0. 2 cos - log 1 - sin ^ 2 Expansion in a trigonometric series gives H2(0) = sin 20 I log2(V2 + 1) + ^ rare tan y 9jt 2159 630 E2(8k^ + 1) / 1 1 12k2_\ ^ ^^^ ^^ ( 1 + •T + ... + -rr r - — 5 I sin 2k0 k=2 (4k2 - l)(4k2 - 4) \ ^ ^^ " 1 Bk-^ + 1/ k=l 62V sin 2k0 (4.53) 92 NACA TM 1324 where Sg = - 0.27412 i 6^ = - 0.08127 ; 5g = - 0.05198 j 6q = - 0.03641, ... The usual theory for determining the circulation OP T(^(-a. cos e) = [_, Bn sin n0 n^l gives the equation E/ jtn\ J ^ sC-a cos e) Bn^l + T~^sin n0 = 2jrca sin 9 \-aa cos 9 - it=l I 4aca 2 = - jtca^a sin 20 - — HgC©) (4.54) from which without difficulty Bsk+l =0 (^ = 0'1'2, ...) Bg = - 0.7304 (xca2 ; B4^ = 0.0047 aca2 ; Bg = 0.0021 crca2 ; Bg =0.0011 aca2, The lift force is found equal to zero and the induced djrag and moment of the forces about the Ox axis are W = 0.4191 pa^c^a* ■' ^ = 0.5737 pa^c^a* (4.55) The first gives an error of 131 percent, the second of 52 percent. By a combination of the obtained solutions it would have been possible to obtain further examples. From the examples given it is clear that for the case of a circular wing considerable deviations are obtained between the usual and the exact theories. Translated by S. Reiss National Advisory Committee for Aeronautics REFERENCES 1. Kinner, W.: Die Kreisformige Tragflache auf potential theoretiscner Grundlage. Ingenieur Archiv, Bd. VIII, 1937, pp. 47-80. 2. Soramerfeld, A.: Uber verzweigte Potentiale im Raum. Proc. Lond. Math. Soc, 1897, pp. 395-429. MCA TM 1324 93 Figure 1. Figure 2 NACA-Langley - 1-5-53 - 1000 CO . > M Q *3 -2 o •a < o T3 C n1 OQ g. >> U n H o < « a a >» iS CO r; a •^ eg a o S o. a 1-. 0. eg ^ DO 03 " rt Qj ed qj .. , r* (- 4) -ti i -a = .« y s w Z Z CO -S 2 . s o> t, ?, e •a c 0) — OQ c >.»■ -c I CM a i OJ c rt 00 •vH 00 !> ■-< S CM O -. 0) • 'i ^-' B^ g.2 o -a " >> s --' •a '' -o 01 * CO 5 0) > a. Ol •o cil O w?? eg — • CO zs «5i2?5 < i3 ?i 2 - aa c " £ 3 •" ■■= »M •r, = C P - o a. 0) c o o ^ - ° 2 ?> ■•5 « S [3 c 3 3s G ■" M 3 ^ ?, " t. J3 = m ° rt M t- C iM Li C 0) o Q, 2 3 bp g " o rt o u u S O.T) c s ■" u t. u -• d > ^ (U c5 S « a 5 u rt " N ^ ;^ u o. . h u o u, >.s £ c 0) -" -a 1. o o tn F 01 £ , , ^ o rt ^^ =!• "i a 6 a E H a o u < z 1 CM 01 < eg eg a OT '-> eg 2,r-i ^ 00 -id B >. Q ^ :::? n >-§ ■a :;5 §< .2 rt S 3 O :S -< ■^2 ,« « ^ r •S <• « 0) !5 ?u2£S; o <; C jj £ ; U5 Z (1. S S . _; aa S » "a 3 -O t. -g H OJ ^ 0) '"tea S gi .22 c o U 0) o)£ -c ,„ ■" o ^ § « -5 - ° t; « <" y, •? i- o "" s m rt V QJ s a> o 2 u) " v S ' ~ .. . . r- 0) ■*i o> ;:: a 111 !« o a. 2 2 0,-a 0) S o £ S3 a «i -u 2 < (J — 0) L.-'S i: -o o o 3 C c ■o = a 4, 2 S S 2 H 6 S - n 01 S £ T> •«. o o ■c „ « °- c * B u I eg 00 ^i eM I eg 0) • ii o .2 "13 ■o 21. S o .J •a c a a g, ZS S"2 5i .13 < ^ 0) 5 ■goa£S! o <; C <5 « , !^ z a, S S *7 ™ 0) "a j:= u ^ ra 4) ■" i: J= ^ -s C o o,£ •" o c o o s o a ?f m < 0) •n o £ nt fc: 0) O — t, 3 " w ^ 0> ^ o. rt 9 -S rt 01 ^ x: ■n ,.^ P o £ ^^ a> o. 0) < < I _ g c a ij J) 13 3550 u " V X > « " CM •3 g c a 3 5S0 u c o o "S 5 u TJ •- " ., a S -o ^ -a i? S 5 " 01 S «i « ' c .;■ rt c o o 2 S 0) c ■s 8 4) •» i ^ "£ o " c « g ■" o c »> S o « ^ S « — g 2 u. 3 d « 9) . ft U ^^ (-) ■3 m O' ** d 5 o a o S " « a 8 " O' -c <: - 2^ it 00 a> 0) r- -^ So ■?, o *j a ^ u u ^ o 0) c^ u Hi _ « t. g o U bo ^ a o • s- 00 -J rt g| O D. ^ S t •< u < z V (30 o » s B ? 01 H ■a . 01 J= 00 A < 13 S3 < 2 X 00 0^ u rt o (0 •g t. o ■^ o* - =i ^ o 0) 2 •S - g S * D. o " t. 8- u S < S n b c o o •o « ■a ^ c3 01 I. T3 0> > £ OJ 2 « - ^■^^ jc ^ Z3 ±; 01 ~ 3 3 00 5 ■'' 00 ^^ 00 r- rt " « .Si J5 -a c « .„ o , c 5 «! ^ « g *' S c « o S.2 .2 DO ^T .-< ^ •-• rt w ''^ ^ -^ "o z: O) c u n «l c s o •«• -g 01 £ g o 01 V 8 « <£ 1! 5. « ^ 0) O II n g-" . e o> « s •;: O M S o T) a 1 " a; 0) c4 « -O T3 ■g t- o t- O- 0) u o. £ £ ?o >> S " .■a 3 0) S ■" 1 -o 2 £ - « Kk - =i ^ o 0) 2 -^ « g o * 0) 2 a 5 u e 00 ^ I » rt 3 .S •a g c a rt Q) at 35S0- u rt 0) 2 § rt * CM rt 3 CT IP" c4 Q) rt 3£So u << I" 3 BuSS2 •o 5 CM (u 2 g ^ I < rt z 2 0) 5 "o o ■D " C ^ 0) <0 " — -• TS ■" -3 is 5 c rt 2 rt c c 5 L. B rt S 9 o -c "I e « rt ♦^ .1, £ o-e o •" -r .a — g i - T3 ? 00 0> 0) " £ g *^ 01 u o> 50-0 « _■ — rt "" g Q. O 19 ^ S c: S ? £ £ •a -- c :2 - « S 00 c 00 E " - -' rt m u ^ 8 « 01 o. *" 00 o bti'5 O) O 5£ S3 CO ■= 5= ffw. ?* a i o ■a 2 - -^ >. u u - 1-r ^ 3 a* r> a> . a> Q, CO rt 0) «?, 00 X3 — 0) „ — (n a> u it .c id 01 5 -a 00 " ■a ° " rt S t3 5 rt 0) rt -O T3 V, ■3 U O O — rt e 0> u o, B u £ - N ■^ ::z bn 01 n .-4 gs * OS'- e S- « 2j a, 00 2 g « OS H I ■ 0) ^ 3 00 5 ■^ o m — ^ 00 r- rt >< rt " ■a .-a c ..J^ o rt c g 0) B u a a o o x: ^ S « °-a.2 •o GO OJ 0) * ££ 5 £ ^ g o. 2 o S " ° rt o> 2 .5 -i -r 00 c :« rt u S ° 2^ = 2 ^ < ^ °!c D- " >- -- , ?* 4) o T3 a, £ "• £ — >> o 0) >. — rt 2^ 0> t- QJ -2 J o -a a «) r- O ^H rt o> '■u 2 I •s < 2 a j= a 3 01 £r; S .^ .^ bD B -2 C o> 2 ■5 0; 0) rt u o. 1 OJ I OJ ° 3 "■a > Li - - .s ■ CO i3 .^ 00 I* »-( « a Si CD ■w CD 2 o - •a 2^ -a 3 2 §< " ll o aS ^ CO _4 • e>j -' ^ CO W C>5 « *■ 1 ■^ < j3 j z ci. S S2^ « a I s < z a „- 2 s H a MM"" 3 2 '^ » s = -a g = ? g-C >,*£ <- S t, o o ^^ <*-■ CQ a 4) Q< o a, a o "I c-i o o >^ "2 «j3 =-,: zs rt 5^S q ^ I a 1 a i! < <3 0) <5 ■g u3£ S^ Sziss2 c g » 5 » S S S o 15 2 -jj ■a »> t^ 3 c 2 c < z > C4 an I M (U ■ -Z Q. "1J > a. •o o J e _o > -ioa ^j- " 2 CO -H 2-3 « c (1) b ■a o. _- 5 a - o o o -= = £ ai t- a 0.0 >>» £ ■° -o .« t, o o « £ »- Oi B CJ Ill" 3||o u rt Q) 2! (11 3 n III" c4 ^ ed u « I" 2 < ■o :^? = 5 (D = i _ o a .■030 ) 5 rj o ' >. :"^ =: ) 00 r- rt ' ^ « ii c - o « c c I >.g H ■ U h A o o j: ;^ S « 3 § ^ c — a a c "i 1 - a n c :;^ 00 ed u 0) ^ (0 *- ?. o a 9 ws.* <- S rt ■2 so" I So *'r--T!Sa'g-H«i 00 a; V a- r- - a> ui o 50-0 r, ft) »< ■= S ■= ■" 5 *- -*^ rt -0) ^ la a; V rt u CO rt 5.'° >. o " .•S 3 a. l-g ° O « rt ■- w S " * a o I" u < < z ■a a a jz ^ ^ - -3 = - •? ^ y y o ■o 5 >. _ c »> ^ * — S m C ™ c ^ o rt c g W b u e ° O g " S a, « ■" 00 c 1 1' T) 5 CJ >^ Ui £ <: ■ 5 s M =* ■C S (u c '^ T3 # :« 01 01 a> a £ c 00 00 c H (1) ei a> u 01 ^ ■0 o. s u. 00 Xf 1 u Si ■a s t-( t-< 00 01 0^ rt u TV CJ 00 01 rt 0) £ £• 0) 5--S S >> o " S! 3 0) o c •" ■3 T3 5 0) 5 "S O 0) j5 ■- ♦.go* a o * t, o> g a ^ u B a " < I u e ■♦ -^ 1 eg > 1 CO T3 H c a c< OJ ct 3 04 en u a> B (0 £ 5 0; •O « J3 "^ — 1> o 00 c « 2 « c S n o £ a — o — JS o ■3 i; 5 « g *' 2 C 0( si « 0> ^ -J 5 >- 3 ■t; 0) a» "00. 5 o « 00 ^ c • •s s •5 s o< * 00 0^ 0) Oj £ c ■V c 00 H s eg o> IS c 01 >. HI E T3 ni n CO < a tj a <: a ■P ss is ^ 01 a; 01 Id P«; 0) "g M O to ■5.OT « .■d -Q T3 ■g I. o t. S- 0< o a c *3 u ^- u u ^ O 00 c ::: CO cd u a> - 01 t- £ o ^ ^ >. U "^^ £ 2o >. o " .■S 3 « u ^ j: 05 — 1 -o 2 *. .* — bD — = ":3 == 0) .2 •- w E " o. o I" t. 01 g a a u c 00 J: ■< u < z I I § I eg CO o .2 "■a ■S CO ™ >• u ::^ 7Z ■a O < •o c 03 5 ' * CO W CO cH *^ I •- <■ « 41 JS f u3 £ S3 g < '^ 5 S S ^ z 0. S S S -T 'O ::^ <« j: u Q) rt " ii -c " c " =.5 c o o ^ 2 5 ° o a; s - - «J c: " (U «--'=-•"' Q *J CO i^ ° rt CO u, O a. S hf S " O ci o to t< w 5 <> a-o c « TJ ■^ . £ ?i = >> ■" u t. t, m 5 o o ■*-» ^ i*^ a> a, -c c 5 IJ c; (U £ c£ C iU 2 a; , 3 — * t- (J a; a> J:; u Q. rt rt a t. ca N D- ^- £ '" CO c n "• 2 S c -S c« si £ 0.2 S •=> 73 ^ t, o o » £ -^ Q. m a. i« : (V a B o I eg 01 I c O. I ■o cd o >> = ^ I S I — ■< J2 <" 2 q <: C n ^, :<: z o< s S ' ^- CO — ' J5 « a, 3 XI (D a Z a t: £ ^ c i c: o ° '" S£ ° O 4, 5 - - "St m T3 T3 — I j3 w ^ x; — CO H« CO Ih 0) O o, o 3 M g " O c« o CO ^ ^ u < 0) -a ■^ £ S £ >■ ■" U ti u CO 3 O O ■fcJ TJ ■-< Ql d 1= -^ ^ » — § 5 ^ — 3 QJ 0) c: i: CO c S « 2 S c:S ■= 2 5 HI f- c a o 2i2 0) ,. 3 t: cu rt £ a> = " u = S a" S u o £5^ ill S 3 cu £ 01 *■ tl^ CD .* £ •a T3 », L. O O « £ - 0. -S a. en = (11 a B u CO ZS B ^ CO CJ « HI • 13 C CO o 5 - o 5 i c« C- s >. Q •a ^.'S ^ CD 2 > b -^ ^ S 5 S S > a, r/} cQ CO T3 cit o 5>-' >B 5r — « S ^2 CD -ti jtf eg c" ^ 'S S = o. ■S < * dj 3 •g o 3 £ S s u: z 0. S S s ■< u < z Us w < cd [-. Z Z M 1 eg en < CM 1 cq » e«] (1) a. P t-l f tj ■2 oo s-i ^ CD ,_) ■ CM — • CO z s « = J - H -a s c CM ? aco M CD o >5 ■Su2£f g o < ^ 2 ii S ^ z o. S S S « 9 S < < z •o £ c a <4 Ml (Q u « 4> 2 •o E C D. rt 4} rt 2 Sf 3 § o u Q> ra "3 »> 5 ■5 3 Q) W "■as c *:^ b S o o 2 B - « S " a) 3 -3 2 '^ 3 o3 « ■a o a !3 « S < < z £a-s£ D3 t. U B OQ ^ B U O c o g 4) T3 u a 00 3 c 3 T3 OQ 13 0, c Ui TJ C V OP > (» m ■a i ■a 00 a u 00 rt (I) u 3 Q. o 0) ■a c c c: u C4 i <4 c 0) B a u a 00 « 0. c 09 0} u ■3 s B x: rt t4 CO c to (U ■0 il> 3 B c x: vi x: x c ^ ■£ 01 H S V a! T3 0} T3 •a CO u B o B T3 .2 OQ c a; •a c a. .a 3 CO c« •a J3 1 ■3 t-i u HI x: aS a (0 v <0 rt 0. •2 -a a" 10 c :;3 in CO cj ID _ CO P.S ° M be ^ >a nl ' T3 g c a (4 (U Rl 3£5o 0.SS2 § ^- cd 1 f= B Q, c« cit 3 5 S <= SS2 B 00 L. B O O B >- o U U OJ CQ • a cj S B CO S o 4; T3 n ■3 4) 5 0) > ii " d •a « 01 B c 8 ■ •a £ ■a 3 ^ 3 m •o o 73 a E w O "SB >. 2 I. Ui B cd o o x: 2 B « cu w x: « g " c^ °* " 2 c 1" S I .2 9t *j zi s « — 2 >- 3 ki .Z; Li « 4> « Sua B o C4 g rt a <1> -w x: «■ g i - ^ r 3 00 4; 4) rt ° 4, £ .5 ■£ "5^ a ■g » 4. < z 4J CD s ~ o » a ■o cd " E 41 >, £ C4 4> V a o -o 5£ S V 0> ei E _ JS O CO •5.05 a CK -Q T3 •3 U O O S £ '^ 2* " ■*^ ■» S -S ■= "2 -a 2 -a - CO E =3 m ctf u 4) _ m k- ? o o. *" sr > >- M bo ^ Q> o S So '■s ■" <« 0.-0 2 ** Qi cu >. o " =13 4) S|£ "^Tl 2 4. «•? £-"►,;, ♦^ *- .^ be — ts -a c o 41 U bO »^ B U E B ■S 2 _o 4) ^ til jB 4> ::3 ■0 ■a > 41 3 00 T3 hi T3 B . > « 00 13 Tl « X3 'oo E rt 00 ■^a i m E E « ■a 4) CD B cii E go CJ 4> £ x: B « 4> rt . Ih a .c Q. 5 ° 4> ki B Ui to — ■a 4) 4) C4 a> x: x: E ^ c x: 10 H-g CO 4; B .2 E 41 I- »;2 s E " 4; L. 4> T3 4) H "5 -E g •< Z c4 3 00 4J jC •a «^ 10 09 41 C4 5 -3 E a? ": " 2 £ « g ■" a 00 2 E « S o ^ 0) 3 -3 2 >- 3 --H Cfl 2 u o, C 00 c :;3 m ol u (U ^ CO p.1 ° Q o ■ *^ "oo ■2 bp« ill a-ss .a 3 4, Sl£ ■S-02 4) cd-a *^ *j — I bo •- C -J B O 4) 2 -^ - a 3 * a o I" u S B g-JS < z I UNIVERSITY OF FLORIDA 3 1262 08105 803 3 UNIVERSITY OF FLOBrn^ \ DOCUIv ,-FUBBAfW 120' J.fo...--'^ • ;,,::'e°a 32611-7011 OSA