NAVY DEPARTMENT THE DAVID W. TAYLOR MODEL BASIN WASHINGTON 7, D.C. THE FORCES AND MOMENTS ACTING ON A BODY MOVING IN AN ARBITRARY POTENTIAL STREAM by William E. Cummins June 1953 Report 780 THE FORCES AND MOMENTS ACTING ON A BODY MOVING IN AN ARBITRARY POTENTIAL STREAM by William E. Cummins June 1953 Report 780 TABLE OF CONTENTS Page ABSTRACT © ovis. Ba ASU DAUM SN Gl ee hl ie a ae OES 7 a 1 INTRODUCTION: s7uretens, cies Selita Wt ces Jans ORT tae ae Selenide Aegan a 1 NSSUMIPSTTIONS) « feiiissesseen ks aeatead phe AN 25 Stata bese) a et lee eat EO oes ip 0) EFFECT OF TRANSLATION OF AXES. «wu... CA cla ei Rol eae A 3 HYDRODYNAMIC ROR CEM settee tee eh as a ae 6 THE“ LAGAWEY FORCE? #Eian.c5).. 5808 Or ternu Meee a cee ete oe 11 THE: HOR CEs wk es eto. pA eS Ae ee eee 18 RHE PORCESRG ce eer onmen hn! | | alent SAUNT NEE C7 ne St Aa 20 HAY OD YI NAMNCEMOMEIN (IOs cece sscs sc. 5 os 0r2 eee eee nce eR eco enh . 22 THE “LAGALLY MOMENT”? M, iwuoeeeis'ains als seagdiswe ecias sees tate souleinn souelaeeeeimecm ncaa a bis einen einet eleleren aeeree 23 THE MOMENT DUE TOMROTATION: Mc cectc cc tee ee ict ce eee 1696 MOWAINGo STNG UZ AIR THR NEGG testes ay secs ll 7 alam ed ltairae es eee ae pea a7 POTENTIAL OF TRE UNDISEWRBED SPREAM (2c cfc cece eee a 3) SOURCESLAND DOUBWEES | 21 ctu nce ECO hae 34 GON CIEUSTON ee enc a ad Gh ae VAL a i 36 ACKNOWLEDGMENTS ete ee are ee 36 APPENDIX 1 - EVALUATION OF INTEGRALS IN F, AND hi, Coo 37 INTEGRA PING (yc slote a Ue CA Ie eee 39 INTE GRAMSHIA(Sy sit i1) 0 Space polar coaedinaces cos 6 Mass density of the fluid Velocity potential of the undisturbed stream Velocity potential due to the presence of the body Net velocity sotential of the flow Potential of the velocity q,, Angular velocity of the body +) , u, +) 3 , AY c Len) n ( Ras t Py ei f is a mt ih i, Mf if Y Oe YP Pais Gy zt Da hie ef Bele ' \ “) _ ALA # : ‘ > * a . \ - ‘ v n PM gery io : oY, y be ee ak ae Ut Pat balay. (i x : < ean ey OR Fes? i ; vi RI oneal ear , Hit e Lee es ahaa A ae e i EIN Sh ele } 7 ° " ' fe al i POR a CEM A SEE See Tn CME A Game BG TLED ong eps 1 1 hs ; : : j STi oN : ; & ; NRHP aU Cah AN Gv dk raga TRE ay Rig . : RIEL BTR E AED ane p a . \ PEA ? 4p Hie Fe . r ¥ y 4 SEINE. Ly \ -. , \ eal > t i ‘ =i ) ; 2 fi a . aa, 3 * * - ; ul ‘ j EES Bue 3 ; DAN eat , A > iS ri ig * we ie as i sab) wi ‘ . ‘ ~ é i 0) » * " rf re ‘ Wea t "Ny i > ABSTRACT The force and moment on a body placed in an arbitrary steady potential flow were found by Lagally when the body can be represented by a system of singularities interior to the surface of the body. They were found to be simple functions of the strengths of the singularities and the character of the undis- turbed stream in the neighborhood of the singularities. In the present paper, - this result is rederived and extended to the case in which the body is subject to an arbitrary non-steady motion (including rotation) in a stream which is chang- ing with time. The force and moment are found to be the ‘‘Lagally force and moment’’ plus additional components. These additional components are given _ for the force as simple functions of the singularities used in establishing the boundary condition and of.the motion of the body, but an integration over the sur- face of the body is required for the moment. INTRODUCTION The determination of the force and moment acting on a body placed in a non-uniform po- tential flow of an ideal fluid has received considerable attention, the problem being of consid- erable importance in both aerodynamic and hydrodynamic applications. There have been two essentially different approaches to the question. In the first, the flow is assumed to be only slightly non-uniform, and the dynamic action is found in terms of virtual mass. Thus the problem is reduced to the case of motion in a uniform stream. Lord Kelvin! solved the important special case of the sphere as early as 1873, but G.I. Taylor’? “was the first to make an extensive study of arbitrary bodies. His analysis, which applied to a steady state system only, included some discussion of the moment. Tollmien developed a solution for the force and moment in terms of the ‘‘Kelvin impulses’”’ and extended the discus- sion of force to include the case of uniform translation in a steady non-uniform stream.* These results have been rederived by Pistolesi > who found an error in Tollmien’s formula for the moment, The second approach considers the boundary condition at the surface of the body to be established by means of a system of singularities within the body. The force and moment are then found in terms of the strengths of the singularities and the character of the basic stream in the neighborhood of the singularities. Hence, this method is not restricted to slightly non- uniform streams, but is limited to those cases in which a suitable system of singularities can be found which simulates the presence of the body. This is the approach used in the present paper. lReferences are listed on page 48. 2 Munk was the first to find the force acting on a body generated by sources.® Lagally apparently solved the problem independently about the same time.’ Since Lagally’s treatment was far more comprehensive and included a discussion of the moment, the statement of the force and moment in terms of the singularities has come to be known as ‘‘Lagally’s Theorem.’’ Glauert applied the method to the study of bodies in a converging stream in order to find a correction for the force on a body when tested in a wind tunnel with a pressure gradient.® Betz derived the force and moment with a somewhat iess mathematicai approach than Lagally and presented the results in a very convenient form.? Mohr discussed distributions of singu- larities over the surface of the body.1° Brard has recently attempted to extend Lagally’s meth- od to unsteady flows but was unable to present formulas of the same simple type as those which hold for the steady state case.!! It is evident that if a singularity distribution is known which establishes the boundary condition, the flow is completely determined, and, in principle, the force and moment can be immediately found by integrations over the surface. However, in addition to possible diffi- culties in performing the integrations, the fact that the pressure is a nonlinear function of the potential is a severe limitation. It is desirable to be able to superimpose known flows to ob- tain new flows and to obtain the resulting force and moment in some simple manner. For steady flows, Lagally’s theorem provides just such a formulation. In the present report, Lagally’s theorem is rederived for genera] singularities, and the analysis is extended to the case of non-steady streams and non-steady motions of the body (ro- tation as well as translation). The force and moment are found to consist of the steady state ‘‘Lagally force and moment’’ plus additional components due to the changing flow. The addi- tional force is stated in simple form in terms of the strengths of the singularities and the mo- tion of the body, but the moment is found to require an integration over the surface of the body. However, in the latter case, the integrand is a linear function of the potential, permitting the Superposition of known flows. ASSUMPTIONS 1. The velocity field is irrotational and has a velocity potential O(a, y, 2, t) 2. If the body were not present, the stream would have a velocity potential 6, which we call the potential of the ‘‘undisturbed stream.”’ 3, There are no singularities of the undisturbed stream in the region occupied by the body. 4. The boundary condition at the surface of the body is satisfied by superimposing a sys- tem of singularities upon the undisturbed stream, such singularities falling within the region which the body would occupy. The potential of the system of singularities is designated by gp, Then =¢+¢, (1] EFFECT OF TRANSLATION OF AXES A point fixed in the body is selected as the origin of a moving system of coordinates; the axes remain parallel to a second system which is fixed in space (see Figure 1). The posi- tion vector of any point of space with 2 respect to the stationary system is designated by r and with respect to the moving systembyr,. The posi- tion vector of the moving originisr,, and its velocity is v,- The following relations hold: (e) m e-f =o, [2] Y= Ye = Ya Cae On im Figure 1 where the subscript m refers to the moving system. The velocity of a fluid particle with respect to the moving origin is related to its abso- lute velocity by i ap SUVs where q,, is the relative velocity and q is the absolute velocity. Since v, is a function of time only, it satisfies the identity Moe ValNe oo) where (7) + oO ) Va Sy Or ewe yee MOeE BT gE ie DY m dz Therefore Gs 7) ae Ye OD) or, since oz, ie Um ‘e 02, an Ox Oy dz we have V_=V and a= V ,(® Savion i) * Hence q,, satisfies a velocity potential which we designate by ®,. It is related to ® by OG Yon Gop Y) = WG AU Upp Gay Y) e875 oT, [3] The pressure at any point, not considering the gravity field and an additive function of time, is 1 a® 1 a® De ee Noes cpm g ala uc alka t Vl 82a [4] From Equation [3] 9 Or, 0a, dv, Or, —= Vo, +-——+—- tr civnee at di Tat) WEEE ait Seana Gale But Oke e dt, ae at dt re so ao oo, av, 2 SG) 0X7, at - ro+VoeVv [5] Bled vale wild OF etal eth tek aaa Substituting in Equation [4] and collecting terms, a® dv 1 m oO p=-+04,°4n+p—"-p—"t [6] in which a term containing v, - v, has been dropped. This is permissible, since v, is a func- tion only of time, and the net force or moment due to a constant pressure acting on a closed surface is zero. If the velocity field q,, were to be considered absolute rather than relative, and the pressure were calculated accordingly, we should have 1 aor Pin Ge @ Gh? Sia at so we can write av De Boar, 1 Up (7] By means of this relation, the flow relative to the moving axes can be considered as if it were the actual flow, and the forces and moments so obtained can be converted to the true values. Thus, for the force exerted on a given surface S, dy Fi= [ Pm nde ~ | p(G2-1,) nde S S t where n is the inwardly directed unit normal to S. By Gauss’ theorem ee ) f(s ) a —2.r |ndo=- —2. a in o - Ue |) Oe s V and since where ¥ is the volume of §. Hence dy F.=F,,+9¥—2 [8] Similarly, for the moment about the origin of the moving system, dv MK, = { pt, xndo = | mtn xndo~p | (Se -1,) r, xndo S S S By Gauss’ theorem again, dv, dv, FF “Th r, Xndo= Mv x F Siifay| fees ar S and we have dv dv dv, dv r|(Fe-r0) “| mae ve ta ee mn)? arr since V x r= 0. Therefore dv, : dv, 4 dv, ) la of NU st) Che x r,, aT = 7 oe Uae V where ls is the centroid of the body relative to the origin of the moving system. Therefore avy Me = Ma +0 dt ) be Thus the problem has been reduced to the case of a body at rest in space or in rotation about some point fixed in space. This is the case which will be considered in the remainder of this paper. HYDRODYNAMIC FORCE We suppose the singularities generating the surface of the body to be enclosed at time t, by a control surface, S$’, which everywhere lies within S (see Figure 2). At present we Figure 2b Figure 2 specify only that S’ possess a clearly defined normal at each point. This control surface is considered fixed with respect to S. The portion of the body between S and S’is designated by V’. The body being in rotation, the space occupied by V “changes with time, and we designate this region by V ‘(¢). In the following discussion we consider the particular set of fluid parti- cles which at time ¢, occupy the region V “(¢,). Since the fluid is also in motion, the region occupied by this set of particles is also a function of time, V (2). By definition then Vt,)= Vat.) but at any other time, in general V (ty )# V(t) The net force acting on this set of fluid particles at any time is d7/dt where ™ is the total momentum of the fluid in V(t). At time ¢,, then F.+F in Ss s dt t oO where F , is the net force acting on § and F, -is the net force acting on $’. Since 1 a® F oH | pndo=—| [ee ° a= 9 nde S S and m =“ oqdr uy we have ae -{ Zola a) nda | 0 8-nao| tf ear] [10] S* Bo. Ge : Vy we) which is precisely the force in which we are interested. At a point fixed with respect to the body, gd _yo.y+98 [11] where v is the velocity of the point. As the origin is supposed stationary, and is also fixed with respect to the body, the point being considered is in the most general case in rotation about the origin, so V=@xr [12] where w is the angular velocity of the body. We have | p @ndo= | p28 ndo+| e(q° wxr)ndo ge ot Ss’ dt Se We can write | ptOnda-2 | e®ndo-- pon ge =e pPndo-w x onda: ge dt dt J, 57 dt dt Jo- 5’ since Then ta | p nde = 4 | pondo-ux | pondo+ | e(q-@xr)Nndo [13] 5% GOs s s The last term of Equation [10], the time derivative of a volume integral whose bounds are changing, must be converted into a more convenient form. At the time ¢, + 52, the initial bounding surface § will still be a bounding surface, but it will have rotated by an amount w@ 5t about the origin. The surface of Vy which coincided with S” at time ¢, will have become some new surface, S’(see Figure 2a). The portion of the body between S and S”’ is desig- nated by V ”. At time Z, Mi(t,) = eq(t jar v(t) At time Ga ot M(t, +50) eq(t, +dt)dr v(t +8t) and sm- | pq(t, + dt)dr =f eq(t,)dr V °° (t+5t) V(t.) The two surfaces S’ and §” are considered fixed with respect to the body. The portion of the body interior to §’ and exterior $” is designated by V, and the portion interior to S” and exterior to S’ by V, (see Figure 2b). Then Vi Vie aie and om.- | pa(t, +8tdr -| eq (t,)dr V(t +t) V (t,) ({ -| | eat, + du)dr Vi (t,t8t) V a(t ,+0t) [14] The velocity of a fluid particle on S’ relative to the body is q+Fx@ Therefore, the normal distance between S’.and §”, the amount the control surface is deformed relative to the body in time 67, is \(qq+rx@)-n| dt Accordingly, in the expression sor dM, we can write for dr dr=-(q+rx@)*ndtdo inV, dt =(q+rx@): Ndtdo inV, where do is taken on §% Then, since the portions of S$’ which bound V, and V, complement each other, (| -| J eat, + dear =F] of ar, + 8619") Vs (t,tdt) V(t, +8t) S“(t,+5e) +(F x we n)a(t, + 8¢)) a¢do Substituting this in Equation [14] and allowing 6¢ to approach zero, we have el padr - | pala n) do ~ [ BE eo Clue [15] Vie ne cis We can further reduce the volume integral which appears in [15], since padr=—o | odr =o | Ondo (16] v’ Vang S+S ” by Gauss’ theorem. The unit normal can be written By Green’s reciprocal theorem 10 since ® is regular through V’. Therefore, since r=ir+jy+ke we have ) | Ondo=po peep || r(q°-n)do e on , Sig S+S S+S But on S we have the boundary condition q:n=wxr-n so { paar = —o | ra xt ndo~pl r(q°n)doa Var s cl The first term on the right can be easily reduced. One form resulting from Gauss’ theorem is, (Reference 12, p. 52) [ om? bao = -/ [acy b+ V yal dr S V Using this, [ho xremdo | (rx w)dt =(" xw)¥ V & S since Ve(rx@)=@°(V xr)—r(V x w) =0 and (rx@-V)r=rxe Therefore, / pqdr =-1, x w0¥—o[ r(g°n)do Va s- Then uch =|- dw ey aes eds : dt | eee =| (1, «4 J+ ww) rw vo zp Bice [17] ial since SS Sp se mH dt & Summarizing, when we combine Equations [10], [13], [15], and [17], we have ee Jee *q)n-(q: maldo- 2 ola on) + on do if S =| afte ma~ (xu gn 6 x w)| do {18] 5’ |r, x Sew, w) +t, (wm: w)| o¥ The first term in the above expression would sive the force if the body were not rotat- ing and the undisturbed stream were steady, i.e., the ‘‘Lagally force.’’ The second term is due to the change of the flow with time, and the last two terms arise when the body is in rota- tion. Since these various components will be discussed separately, we call them F,, F,, F,, respectively. = =| of 5c -q)n—-(q- na ds: [19a] Sale d ’ Be ele ee ein Ge oN [19¢] -[r, xau_s (r, e w)+T (o> w)|o# If the origin of the system of axes is taken to coincide with the centroid of the body, the last term of the expression for F, vanishes. The above forces are defined in terms of integrations over the control surface S’. Since S“ has not been specified, it is evident that the forces are independent of the particular choice of S’, as long as it satisfies the conditions necessary for the integrations to be carried out. \ THE ‘‘LAGALLY FORCE”, F, Initially, we suppose the singularities generating the body to be discrete, isolated, and fixed with respect to the body. Their locations are designated by the set of position vectors r,. For the control surface, we select a set of spheres S, with their respective centers at the 12 singularities and their radii &, chosen sufficiently small so that no.two spheres overlap. We designate by F, (i) the integral in F. evaluated over the sphere $,. Then F,=2F,@ [20] Since F, is independent of the particular choice of 4, F,(<) is independent of F,. We refer the region around the singularity at r, to a system of space polar coordinates with the origin at r, (see Figure 3). x—-2,=f cos 0 [21a] y-y,=F sin 6 cos X (21b] 2-2, =R sin @ sin X [21c] Figure 3 The quantities appearing in Equation [19a] may be written n=icos §+j sin @cos\+k sin sina [22a] --1[ro,n+0 1s Cattery nat OO SG rSA [23] iL peep on R sin? 0 do = R2 sin Od@dX [25] 13 where o® P= ‘ Do = - oR a0 OX and Sg and s) are the unit vectors in the (# = const., A = const.) and (F = const., @ = const.) directions respectively: $9 =-i sin 0 +j cos @ cos ) +k cos @ sind [22b] S) = —j sin » +k cos d ‘(22c] The components of the force parallel to the i, j, k, directions become: Si 2 2 Cos OB ay2 F,,(@)=— a [- 2,4)? cos 0+ 02 es Oe oy a sin? 6 [26a] +2R,®, Op, sin o| sin 6d@dd 9 27 7 ry) -5| i - (R;®,)? sin? @ cos \ + o sin? @ cos \ + oR cos ) [26b] -2R.® ® sin @cos 6cosA+2RkR.0 0 sinA|dé@dx GO Tid 2) u n \ p 27 pT Re. 0-51 | (#50, sin? @sind + 0g sin?9 sin A+ 0? sin d [26c] ~ 2k, 2, Oy sin@ cos @sindX — AGO 2 cos sjaaan In the regiono 0. The functions of ® which appear in Equation [26] are: co n R®, = > > nR” Ps (a° cos sd + 0S sin s }) n=O S=0O E [28a] oh ; S (n+ IR ™1) PS(aS cos sd + 6 sin sh) n=o Ss=O co n dPs ¢ Oy =- S S a 7 (a; cos sA + b> sins\)sin@ n=O s= 0 B as n Pps [28b] - y y R71)" 2 (G8 cos sd + BS sin sA)sin 9 du n n n=0 SsS=o 9 n ®) =- sR” P* (aS sin sd — 6° cos s\) n=O s=1 [28c] ree = ; S -(n41 sre: x sk ~\" Yps(as sin sd - 6° cos si) n=o s=l1 * When these are substituted in Equation [26], the resulting expressions become quite cumber- some. However, since F(t) is known to be independent of Ff, it is evident that the net co- efficient of fee must vanish unless ¢ = 0, so only the latter terms need be considered. A fur- ther reduction can be made by taking account of the integrations with respect to A since all terms contain products of the type 15 For F(z), a term must vanish unless this product is of the form sin? sd or cos? sd. For the other two components, there is an additional factor, cos\ or sind. Those terms with cos vanish unless the above product is of the form sinsA sin(s+1)A or cossAcos(s+1)A. Those terms containing sind vanish unless the product is sin sAcos(s+#1)A or coss\sin(s+1)X. The components can now be reduced to: Fi, =7p — ; = 5 oe Ge vr bad of |e tle PP Rag 1p Og COE n+1 du ai (Le Se Past Pn Ge [29a] ahj29 dPs ee Bees Pala 42)|dy p p. co n—-1 oP y y = ot Fy (2) = = (6) (GayGnae iia nn nO) anys +l) n=O |.S=0O [29b] n + S n(s)(as thas + bstt BS) [x (el lens) sS=0 co n-1 . _ 7P 5 : 5 ‘ = Fy ,() “3 n(s)(a; +1 ae Gi Os an” *)1,(8, s+ 1) n=0oO sS=0O [29c] = y m(S\(G2., Os —b2 9 a*\ Tis + 1, 3) where 7A) Se. GAS) Thy s>0 16 and ‘ p pl 2 2 nas kK in(e, p+ 1)= | ae ane [= 42) (nV? + PLY F(a + DQP4D elt apPt! pty ape, +(n+1)(1 - p?) u(e? fn Pp lee <1 +1 du n dp P pHi A dP aay n a= we} Gyn du du V Ue The convention is adopted in Equation [29] that bo = 6° =0 n These integrals are evaluated in Appendix 1. The components of F, become P= 2np SS nisy(ah,, a + 08, peace n=O s=0O (ae [30a] (—°) n-1 PiyO=—ne S| DS” aisles ae" + s,, e)@ee sD n=o s=o (n-~s-1) n [3 0b] -> n(s)(azsh ay + bytt 6,\@+ e+ 2)! (n-s)! co n-1 F..(i) =-7e S ) 8 pst igs zs+t\ (nas + 1)! 12(2) = , n( (a, sri “i n+1 a, @=25=0 ae 1)! [30c] eM enn Ga Brea, ae (n- ss)! These components must be evaluated at each singularity. Ff, is then given by Equation [20] The Equations [30] are bilinear forms in the coefficients of the expansion of the total potential, excluding the singularity, and coefficients of the expansion due to the singularity The bilinear character of these expressions has a number of important consequences 17 1. A singularity can be considered to be composed of a number of superimposed singulari- ties, (2, ), (i, ), (¢,) «+, and the forces F,(2, ), F, (2, ), F,(,) +++ determined independently. Then FL@=F)@)+F,G)+F,@G) + - 2. Similarly, the potential excluding the singularity can be considered to be composed of a number of superimposed potentials, and the force due to the interference of each of these with the singularity can be determined separately and Fi@) found by addition. 3. Consider the net force on the body due to the mutual interference of two of the singu- larities within S. By 2, these forces can be determined without consideration of the effects due to all other components of the flow. Instead of evaluating these forces separately Over the spheres S, and Sis let the integrals be taken over a larger sphere Si with its center at r, and R > Lie . The combined potential may be expanded in a form such as Equation [27] which will be convergent for R > Ir; -r ;|- However, since the combined potential must vanish at infinity, all of the unbarred coefficients must be zero. Since the integrals will have pre- cisely the same form as Equation [29], the components must be zero due to the bilinear nature of Equation [80]. 4. In evaluating Equation [30], the unbarred coefficients may be determined for f,» the potential of the undisturbed stream only, rather than the total potential excluding the singu- larity at r,, since by 3 the net force due to the mutual interference of all the body generating singularities is zero. 5. In the case of continuous distributions, we may suppose the region over which the singularities are distributed to be subdivided into small elements. The net potential Ad, oi the portion of distribution within the element A,7, containing the point r, can be written i) n Aid 5 = s s R{"*) Ps (u) (aS cos sd + BS sin sd) A,T n=O s=0oO which converges for all , greater than the maximum distance from the point r, to the bounds of A,r. This has the form of an isolated singularity at r;- Hence Equation [20] can be written F,= » POEs Oo Gao BR) We t If the number of elements is increased indefinitely, the dimensions of each approaching zero, —s Ty . . . then the coefficients a , B> will in general approach limits, and the sum becomes an inte- eral, F, =[F, (O(a, 0, a3, Bg)dr [20’] 18 The corresponding formulas for line distributions and surface distributions are immediately evident. THE FORCE F, The same general procedure is followed for F, as for F.- We again consider S” to be composed of a set of spheres surrounding the singularities, and define F,()=-£ | ofrg-m) +0n) do [31] so F, = 2F,@ [32] If we make the substitution r=r,+ R [33] [31] becomes F() =- p+ | Raa-nydo+t, | @- mao + | Ondo dt |-s S, Sh u L Remembering that these integrals are independent of f,, it can be seen from Equations [23], [25], and [28] that | R(q-n)do SF can involve only the coefficients aa, at @. These are the strengths of doublets with their Dl axes respectively parallel to the x, y, zg axes. The potentials of these doublets are a2 cos @ a!sin@cosA 6! sin @sindA 1 1 f Pole R2 2 R2 If we regard these coefficients as the components of a vector, this vector will have the direc- tion of a single doublet equivalent to the three doublets, and its magnitude will be the strength of this ‘‘resultant’’ doublet. We designate this vector by A, and call it the vector doublet strength of the singularity. The potential and velocity field of a doublet in terms of its vector strength may be written (A) =A a= Ae [34] 19 and (A) = 25 (3(A - myn ~ Al = 3(A + RR _ =e A R? R3 where n has the same meaning as in Equation [22a]. Then i Rq- mde =| 2A Rdo = 2) (A-R)ndo 5; S; & Ri Js; and by Gauss’ theorem, remembering that n is directed outward | pea mao -2/ V(A+ R)dr Sj o V; -8n4 [36] Similarly, [ ®ndo depends only upon A, so S. t [ ondo-4 [ (A-Ryndo=42A (37] S R3 vs 3 i t i by Equation [34]. The remaining ene | q°ndo depends only upon a and is simply the total flow from a source of strength a@°, so : [aendo= sna, [38] Sj : adr. Then, since re =-9;x@ a da? F.(i) = -479 | -@ (rf, x @) +1; Toe ah [39] The extension to continuous distributions is evident: —o Fy=~4np [| -age; xe) +152 dal adr {391 t 20 THE FORCE F, We define F,@--| olrxw-nq-(rxw-q)n+ O(nxw)\lda [40] sf Then FF o-[t, w < Ande = | 2 5 V(R- wx A)dr U V -{ (oxA-V )tdr V; -| (w x A)dr Ve u areas lew x A) 3 L and | (Rxw- q)nds-=-47(w x A) Si 3 L so [ p [exw mg~ (xu @nldo=Se lw q)(r x n) —(q- n)(r x @| do [54a] Ci mM, =-2[( polrxn)do [54b] 2 TUG Nae || pon xr mle xg) + (tw A(t xn) + Olu x xmypdo [54c] 52 While the component M, cannot be reduced, it is a linear function of ©, allowing the superposition of solutions. It should be noted that the components M> M,. M, do not corres- pond exactly to the forces F,, F,, F, since the integral for the force corresponding to M, was broken up into two parts, one becoming part of F, and the other part of F,. THE “LAGALLY MOMENT”, M, % We again suppose the singularities to be discrete and isolated. The moment M, (2) is then mo-{ ofa apie xn) ~ (a: mir x a) do [55] Ss; 24 and M, == M,@ [56] Making the substitution [33], we have M,(i) =", x Fa+{ | 34 > q)(R x mn) —(q- n)(R x | do Ss; [57] =",x F@-| p(q-n)(R xq)do 5; since Rx n=0. Again using polar coordinates, 277 pT M,,(@%) = (, x F,@- i) =| | p R70) sin 6d6@dX [58a] oO Oo 27 pa Myy@) = (r, x F.@)- j) f I pR? ey Pn sin 9 sin \ [5b] + ®)®, cos 9 cos \] ddd 27 rT M, ,(2) = (t, x F,@)- k) | | p Bale Dy®, sin @ cos d ete [58c] + ®) ©, cos 6 sin Ald Oar The same procedure used in obtaining F(z) is followed. The integrals in the above expres- Sions then become 27 pi oo) n 2 , S a Re ip ( pR 2@_) sin 6d0dr=70 (a5 be - be as ) n=] s=1 1 | (2n + 1) (PS)? dp =1 [59a] 25 27 pT i pk? (Op, sin Osind +), cos 6 cos \)dddX oO 00 n-1 7p 2 a oe ee >. n(s)(a,71 6, - by tan) J,(s +1, 8) [59b] n=1 S=0O n-1 ar n(s) (a5 os*? fa bs as) JA(8; 34 1) s=O 27 pt J { oR? (-O6@, sin 9cosA +) ©, cos @ sin A) dedn °° n—-1 — S n(s) (as *} as 4 B*155 ) a (s + Il, 8) [59c] n=1 Ce) n~1 = y n(s) (aS a8t1 4 bs St!) J, (8, 8 + 1) s=0 where n(s) has the same meaning as before, and 1 dpe [ps J,(8, 8£1) = + |nP* 7 -(n+ P+ er = Me du + [nls £1) + s(n +] PS ps*1 4 —* ee Evaluating these integrals, (see Appendix 1), we find that te) n M, (2) = E x F,(i)- i - 2p , > (a5 BS - 68 aS )s ee [60a] n=1 s=1 M, ,(2) = E xF,@- i] +e y S n(s)(ae*t bs 4 a8 a a (60b] _ pti gs)(n+ s+! n — bs qgstl Bie Daisey 26 2) n—-1 My (= [tex Fy] + wp y S n(ai(as*t a n=1 s=o Ts “ Ts+1 ! + pst bog Gt = aE A ——, (n- 8-1)! [60c] The total moment is then given by Equation [56]. Since the expression for M,@) is a bilinear form of the same type as that for F,() the discussion of the latter applies equally well to the moment. Hence, for continuous distri- bution K, =f ROG bs, aS, BS dt [564 THE MOMENT DUE TO ROTATION,M, We define the moment M,(z) to be M, (2) -{ ow xren)(rxq)t+(r xm? qy(r xn) + ®[w x ( xn) edo [61] Se and == M,() [62] The first two terms in the integrand can be reduced as a triple vector product, (rx w- n)(r xq) —(r x wo? Qv(r xn) =r x([(F x ww) x (qx n)] = (x) (r > gq xn) [63] since r> rx @=0. Making use of substitution [33], (rx wo) (t> q xn) =(r, x w)(r;- qxn)+(R xw)(r,>q xn) [64] But by Equation [43], | | (x ole, ax mdo= te, xe) { q xndo =0 [ 65] S; S; Also, since it is evident that | (R x w)(r; > q x n)dz- involves only A, we have, using S. U Equation [35] 27 il (R x w)(r;° axn)do--{ (R xe) (rr; - Axn) 22 : S. RB S; i t -lw «| (r;xA+R)ndo BR; Si - Au x[ V (r; x A+ R)dr V. u ~Anlw x (r; x A)] [66] 3 Using Equation [33], the last term of the integral in Equation [61] becomes | ®lw x (rx n)}do-=| ®[w x (r,x n)] do =@ (1 <{ ondo) S; S; S; since R xn =0. Using Equation [37], we have | ®[w x (r x n)]do = 47 [w x (tr, x A)] [67] S; 3 Substituting these results in Equation [61], we have M, (4) = 0 [68] MOVING SINGULARITIES The cases which have been discussed so far are (1) discrete singularities which are fixed with respect to the body, and (2) continuous distributions of singularities. While these cases include the most important applications, flows exist which can be discussed in terms of discrete singularities moving within the body. In the present section, the analysis will be ex- tended to include this case. The control surface S, enclosing the moving singularity is taken to be a sphere with center fixed at r,(¢,}, the instantaneous position of the singularity at time ¢,. At the time t, + 6t, the singularity will have moved tor,(¢, + 5¢) or referred to the center of the control sphere R, (67). 28 Let the coefficients for the expansion of the potential due to the singularity about r,(¢) be a*, 6°. This potential may also be expressed as an expansion about r,(¢, ) which will con- verge for all |R| > |R,(5¢)|. Let the coefficients of this expansion be a> , 6) . The latter expansion is precisely of the form due to a singularity fixed at r,(¢,). If we find a= , 6% in terms of as, Ge, we may insert the values directly into the formulas for the force and moment. It is evident that PUR AEC). & G.)2 BS (t.) [69] Therefore, the formulas for the Lagally force and moment, which depend only upon the instan- taneous values of the coefficients, remain unchanged. Further, it is only necessary to deter- mine a° and A’, the source and doublet strength of the equivalent singularity, since the time derivatives of these quantities appear in the expression for F, (z) but no higher order terms appear. The potential about r,(¢) may be written @ A- [R -R,(62)] ht Ee R-R,OOl iR-R wo + terms of higher order or, $i 1/2 3/2 : 2 >-R-A: . — je ee R, Re) 4 R-A Se R, ) cs R R2 R3 R2 Expanding by the binomial theorem and collecting terms, we have fo Pr(Aes a°R,) - R + terms of higher order RR? Therefore a “(t) = a%(t) [70] and A’=A+@R [71] 29 Differentiating, At time ts this becomes adh’ dt [el ao on a ie | . [72] where v, is the velocity of the singularity relative to the body. Equation [39] then becomes 5 Xo) da° dA F,()=-400 [ae -4, x0) +8, <2 2h | (73] POTENTIAL OF THE UNDISTURBED STREAM, ¢, It has been seen that the coefficients a°, 6° need be determined only for ¢, the poten- tial of the undisturbed stream. In general, these coefficients can be found in terms of the po- tential and its derivatives at point r,;. Since ¢ is analytic in the neighborhood of r,, it can be expanded in a Taylor’s series about r,. ie dlr) = d(r,) +R -V )d(r;) ei (R-V)? db (r;) veal -v)3 P(E ;) + o+ where R-V= i = gop La - Cn (a =, ) ( ae male Oy a ae = (cos 6 2. +sin 6 cos \ 2 +sin @ sind 2) R(n- V) Ox oy 021 Hence, the expansion can be written p(r) = y = R™ (n-V )"A(r,) [74] Equating coefficients of R” in this expansion and the expansion of the potential in terms of Spherical harmonics, we obtain the system of identities 30 n iL (n- Vv)" d(r;) = > Pi(cos @)(a; cos sd + 68 sin 8A) [75] ni ie s=0O which permit the determination of a=, 5°. The solutions are most conveniently found in the form of recurrence formulas. Since [75] are identities, and the a>, 6° have explicit values in terms of the derivatives of 6, we can write = — (n-v ntl g = oe (cos @) (a7, cos sd + ome sin s i) n eae y P (cos 0@)[cos sd\(n -V Jae + sinsaA (n-V) 6°] n+1 s=0 (76] When the operations are carried out in tlis second form, it can be reduced to a sum of terms of the type A, cos sd, B, sin sd, which are linearly independent, so we may equate coefficients of the two forms. We have then the further system of identities, é asi gps oa (e) Ps ; EN eeu D/2A2 (KOS 6) Res 45 (3 Dey asim\g) | aa n+1 an 2(n+1) n ae n Be da [7a] Ot S+1 aosth EperistnG| a son el )] Oy Oy Ss Pau aoe 1_[opscosoo + 2 ef —_———— B ae 7(s — 1) sind a a 8 ~ XGp a 1) F) st] a pstt repos sina ( atid payee ) 02 oy] - -1 das = 00% dz oy [77b] The special case in which s = n + 1 is easily solved, 1 da” Oe n+1 qttl EAE eth n joe sin of a) nti ntl ~ 50, gy Yin Sf | we 31 and since Livi Qn + 2)! ae oe ayee © = (2n + 1) P" sing J a 0.6” n+l _ eda el OD Re (none py 2) an41 (2n + 1)(2n + 2) t Oy 02 [78a] Similarly gam 9b” prt = SUE cn)(22s n) Dol Gyan ey | ea wep [78b] The special case, s = 0, is also easily solved by setting 6=0. Since Pee (cos 0) = 1, (79] The identities of [77a] and [77b] can be transformed by means of the recurrence formula (see Reference 13, page 360) cos 6 P> = P*, — (n+ 8) sin 6 pest [80] We then have (adigy da; Pn sae pel Oe “a | ; a —1 abst gas La {rs sino| nis - 0 (%2 On J Foe) 2 | mOGE 1) ay dz Or [81a] oastl a bstl Ucn sina ( Tey pte ) dy 02 i Sn Cha a be POWMGE R= =) on). pr sino |qta ~10(22 lk | -2ta+ 0 | aL \ ia Caen) Aye ab) he 02 oy dx [81b] das gos a siaeisim of 1 s-1 S-] da, eT) (2% nes ) [83a] dx (n+S8) Oy Oz By Equation [78a] das nls - 1) (es A are | _s-2) (ome : uo de (2s-1)2s \dxdy dx dz iS Oy 02 Assume Equation [83a] to hold for da" Nadal Figs TY (a 2 a) dx (m+s) Oy dz Then JO a Ona | oS) (a aon) Ox (m+ 8s +1) BP (m + 8)(m +s +1) dx ay axdz Panes) (2¢ns1 2bnt1 (m+s+1)\ oy 02 Therefore, by induction, Equation [83a] holds. Similarly, 0 bs i - Bee eA ee 0 de (n+s) 02 dy 33 Substituting these results in Equation [81], we have eo) feo 2 vn). ty eh aas da Woe ree Une a 2(n + 1) neg eine Vases) De [84a] 1 F) S+1 obst Ape dit sin 9 (“Cn ig atue ) Oy 02 1 1 RO = abs OS Hho ae “a ap 6 n Bell n+l Tel *) RI ae ae oe [84b] s Ss + P+! sin a(- oon. —) B 02 Oy The associated Legendre function P* is of the form c sin*@ f(cos @) in which f€cos 6) has the property that f(cos 0) # 0 Hence, we can divide Equation [84] through by sin*@ and set @ equal to zero to find Once We make use of the recurrence relation 2s cos 6 PS = sin 6 Pet + (n+ s)(n-s+ 1) sin 0 Rem [85] Then Ss +1 ps1 cos o(—2 ) = J sin? (2 ). tal a es) ——_ } sin’ @/ 2s sinS*t! 6 28 sins lg from which ( Pe ee oe d (n + 8)! [86] sin’ 6/9 9 O sin’ 16/4 4 2°s\(n- 8)! 34 since Po(1)= 1. Using this result in Equation [84], the relations [82] follow. Hence, by induction, these relations hold for all n. By Equations [78] and [82], a° n and 67 may be easily evaluated. Since a? = d(r,) and 6° = 0, a; and 6° can be found by repeated use of Equation [78]. Then Equation [82] can be used to find a and 0°. The values of the coefficients a*, 6° are tabulated below for s <4. ye en Cae 5° =0 n n' aa” n n—1 n-1 ai-_?_ 9 _(¢,) eee es) (n+l)! da" (n+ 1)! da") n-2 4n—2 an = 2 2 (pyy — P22) br _———_ = (2¢,2) (n+ 2)! aa"? (n + 2)! axe"? -3 GS eee ES 3¢ ) p3 = 2 a” (34 fe “ (@nb oae® ee Hi OG sa Cae Be n—4 n—4 a esas AGE A Biches NN 5S ae ee OI (C48, A ne ayicget = ee Oe Ua Teams eam tee A SOURCES AND DOUBLETS [87a] [87b] [87c] [87d] - 44222) [87e] When the singularity at r, is a source or doublet, the expressions for the force and mo- ment take particularly simple forms. Using the values for the coefficients given by Equation [87] we have the source FO = 4p ae ee | i Or i _, {a Fy) = 40a oo] t —,[9¢ TAO) = Sg Ca are L F(t) = -479 494, (r;) [88a] [88b] 385 where q .(r,) is the velocity at r, due to the undisturbed stream. Also ~, da? F,(i) =-47p| a3 (r; xw) +1; a [89] t : [90] M,(@)=0,x FL@) For the doublet OAM OTE. a= WOcd GA) 8 Ee al 1 F,si)= Ane] a eau il Ae ee 1 AOE [91a] we slate BEI Ags — a F,,(@) = 470 ar @ +a) a bt 2 {91b] y ae ady | dy? dy dz 2 2 a2 Fei) ~ do a? gee + ai ors. +6 as [91c] which can be conveniently written in vector form F (i) =~4 np (A v4, | “ Also eh dA F(t) = -4 Cae [92] : M, (= (r; x Fi@) >i) + tno (at “. b} **) [93a] ‘, Saar Ty Opiates My y=, x FO j) + 4 no( 34 ae = [93b] , gp re) dd Og M,,@)=(t, x F,@)- k) + Ano (aj oe se) [93c] or M,@)=",xF,@ +470(q, x A); (93 4 36 CONCLUSION We have shown how the force and moment acting on a body with an arbitrary motion through a fluid subject to a time varying potential flow can be found if the body can be repre- sented by a system of singularities placed within the body. The force can be considered to consist of three components. The first, which would be the total force if the instantaneous flow were steady, is simply the ‘‘Lagally force.’’ This is found in terms of general singularities (Equations [20] and [30]). The second component de- pends upon the change with time of the singularity system generating the surface of the body. This force (Equations [32] and [89]) is found to be a function of the strength and orientation of the sources and doublets in the singularity system but not of the higher order singularities. The third component is the force which wouid be required to generate the given motion of the body in a vacuum, if the body were to have the same density as the fluid (Equations [8] and [48]). The moment similarly consists of the ‘‘Lagally moment’’ (Equations [56] and [60]) and additional components, but it has not been possible to resolve these additional moments in the same manner as for the force. They consist of two terms; the first, appearing in Equation [9], is simple enough, but the second requires the evaluation of a surface integral (Equation [54b]). However, the integrand is linear, so it is permissible to superimpose potential flows which satisfy the boundary conditions. ACKNOWLEDGMENTS The assistance of Mrs. Alice Thorpe has been of great value in the preparation of this report. She aided in the reduction of the many integrals and carefully checked the algebraic operations, greatly increasing the author’s confidence in the results. The author is also very grateful to Mr. P. Eisenberg and Mr. M. Tulin for their careful review of the report. 37 APPENDIX 1 EVALUATION OF INTEGRALS IN F, AND M, REFERENCE FORMULAS The associated Legendre functions satisfy certain difference relations which are tabu- lated here for reference (Bateman, Reference 13, p. 360). Ga Be Wi So ce nies (ae ex = 0 [94] V1 = 12 Pst! = 28y PS - (n+ 8)(n—8 +1) J1-p2 PS [95] PS, =uPS+(n—s 41) J1— 2 ps [96] PS, =uPS + (n+ s)\V/1- p?Ps [97] V1 = 22 P+! = (n+ 8 + 1)pPS - (n- 8+ 1)P8,, [98] (ue ee (MnP! = Gas el) Pe [99] (1 =p?) : Z=(n+s)P>_, — uP), m We shall also need the following integrals: “ S (2) (ns & Gee oh oe = n=o |s=o (n— 3\— 1} D (30b] +1 2)! ws nto) (a pon Oo yt eae Py O= ee s 5 "roi (e n+1 bee = OF 5 a?) fe a “= - [30c] 3s Bl = 2)! i D100 (004 Bh Ont s)oesam) where H@)=28 7@ye to os 0 44 Force due to Changing Flow Fz =F nf dae Fi ()=-47p | -a9 (t,x @) +8, a Ge F, ~-|",% amg -w) +t (w «| ey Lagally Moment M,=2 Mw) M,,() = E xF,@): i|- 270 > > (as bs — 68 a: )s Hae ) n—-1 M,,() = E x F@- i + 7p y S n(s)(an** BS +,a$ bS*1 n=l s=o . -os ae Sip as) (m+ Shitaely)! TE (aie! co n—-l My .@)=[rx FO: kK] +p 9S” ncoilont at n=1 s=o Sl FS 4 7 pe! — po For @e se iy 2 UD On Ge Ge bt b? ) (n- 8-1)! Moment due to Changing Flow d M, =— O(r xn) d 2 ZL F - Singularity Moving with Respect to Body a da? F,()=—4n0 [acty, -T; xw) +1; ae 2h) [32] [39] [48] [56] [60a] (60b] [60c] [54b] [73] 45 Coefficients in Expansion for Undisturbed Stream Potential Bae Odo Pefctae eslulias sae erin EY ada a om a) 7 mel” Os iyOp soy oem ea teal n n+1_ 1 0 an 06, fe 2 n+l (2n + 1) (2n + 2) 02 oy (78b] at ih i da, m+Hl mst+1 oa [82a] m=t bt a 1 d br, Ba ae Owl. OB [82b] and in particular n 1° ¢ ; ap = Pi aa” b° =0 [87a] n—-1 n—-1 ES rai ) (gy) Cie eee ae ) [87b] (n+ 1)! da"! (n+1)! da?! n-2 n—2 a-_2 49 (g,,-¢,,) (Se ee eo) [s7c] (n+ 2)! da"? (n + 2)! oa"—2 q Sh es oa Aa MeiOa enor eye) 2 a, G bi 3)! aa%3 (dy yy byzz) oe = @ i 3)! 32°83 (346 yz Doon) (87d] 4 2 a” eae? La i (n+ 4)! aa? rxxx ~8Pyy22) |; @e at 6a Py 772) Syz22) (8Ke) Source F, (i) = -47p a8q, (;) (884 where q.(r,) is the velocity at r, due to the undisturbed stream 46 : = da? F,(¢) = tno | 2, xw) +0; 2 [89] dt M,@)=";xF,@ [90] Doublet F(i)=~4 np | (A 7a.) [917] ee dA Fl) --4 1p Ee) M,@ =", x F,@) +470(q, xA); [937] 47 REFERENCES 1. Kelvin, Lord, **On the Motion of Rigid Solids in a Liquid Circulating Irrotationally Through Perforations in Them or a Fixed Solid,’’ Phil. Mag., 1873, Vol. 45. 2. Taylor, G.I., ‘““The Force Acting on a Body Placed in a Curved and Converging Stream of Fluid,’ ARC R & M 1166, 1928-1929. 3. Taylor, G.I., “‘The Forces on a Body Placed in a Curved or Converging Stream of Fluid,’’ Proc. Roy. Soc. London, Series A, 1928, Vol. 120. 4, Tollmien, W., ‘“‘Uber Krafte und Momente in schwach gekriimmten oder konvergenten Stromungen,’’ Ing. Arch., V. 9, 1938, (Trans. by Stevens Inst., ETT Rep. 363, September 1950). 5. Pistolesi, E., “‘Forze e momenti in una corrente leggermente curva convergente,”’ Commentations, Pont. Acad. Sci., 1944, Vol. 8. 6. Munk, M., ‘‘Some New Aerodynamical Relations,’? NACA Rep. 114, 1921. 7. Lagally, M., ‘““Berechnung der Krafte une Momente, die stromende Flissigkeiten auf ihre Begrenzung ausuben,’’ ZAMM, 1922, Vol. 2. 8. Glauert, H, “‘The Effect of the Static Pressure Gradient on the Drag of a Body Tested in a Wind Tunnel,’’ ARC R & M 1158, 1928-1929. 9. Betz, A., ‘‘Singularitatenverfahren zur Ermittlung der Krafte und Momente aur Korper in Potentialstromungen,’’ Ing. Arch. 1932, Vol. 3 (TMB Translation 241) 10. Mohr, E., ‘“‘Uber die Krafte und Momente, welche Singularitaten auf eine stationare Flussigkeitsstromung ubertragen,’’ Journal fur die reine und angewandte Mathematik, (Crelle’s Jour.) 1940, Vol. 182. 11. Brard, R., ‘‘Cas d’Equivalence entre Carenes et Distributions de Sources et de Puits,”’ Bull. l’Assoc. Tech. Mar. et Aero., 1950, Vol. 49. 12. Milne-Thomson, L.M., ““Theoretical Hydrodynamics,’’ Second Ed., Macmillan Co., New York, 1950. 13, Bateman, H., “Partial Differential Equations of Mathematical Physics,’’ University Press, Cambridge, England, 1932. iD i le ik vs Polvaten. tae Wh i si th pe, ieiepaioi Copies 17 = me of 49 INITIAL DISTRIBUTION Chief, Bureau of Ships, Technical Library (Code 327), for distribution: Technical Library Assistant to Chief (Code 101) Civilian Consultant to Chief (Code 106) Research and Development (Code 300) Applied Science (Code 370) Ship Design (Code 410) Preliminary Design and Ship Protection (Code 420) Preliminary Design (Code 421) Model Basin Liaison (Code 422) Submarines (Code 515) Minesweeping (Code 520) Torpedo Countermeasures (Code 5201) BPerPRPrPNO PEP ee ox Chief, Bureau of Ordnance, Underwater Ordnance, for distribution: 2 Code Re6 2 Code Re3 Chief, Bureau of Aeronautics, Aero and Hydrodynamics Branch Chief of Naval Research, for distribution: — 3 Fluid Mechanics Branch (Code N 426) 1 Undersea Warfare Division (Code 466) 1 Mathematics Branch (Code 432) 1 Naval Sciences Division (Code 460) Director, Office of Naval Research, Branch Office, 346 Broadway, New York 13, N.Y. 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Korvin-Kroukovsky 2 Director of Aeronautical Research, National Advisory Committee for Aeronautics, 1724 F Street, N.W., Washington 25, D.C. att Director, Hydrodynamics Laboratory, Department of Civil and Sanitary Engineer- ing, Massachusetts Institute of Technology, Cambridge 39, Mass. 1 Director, Experimental Naval Tank, Department of Naval Architecture and Marine Engineering, University of Michigan, Ann Arbor, Mich. 1 Director, Institute for Fluid Dynamics and Applied Mathematics, University of Maryland, College Park, Md. il Director, Institute of Aeronautical Sciences, 2 East 64th Street, New York 21, N.Y. 1 Director, Hydraulics Laboratory, Colorado University, Boulder, Colo. 1 Director, Hydraulic Research Laboratory, University of Conn. 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Copies 52 Director of Research, The Technological Institute, Northwestern University, Evanston, Ill. Head, Department of Naval Architecture and Marine Engineering, Massachusetts Institute of Technology, Cambridge 39, Mass. Head, Aeronautical Engineering Department, Catholic University, Washington, D.C, Head, Department of Aeronautical Engineering, Johns Hopkins University, Balti- more 18, Maryland Head, Department of Aeronautical Engineering, Pennsylvania State College, State College, State College, Pa. Head, Department of Aeronautical Engineering and Applied Mechanics, Polytechnic Institute of Brooklyn, 99 Livingston St., Brooklyn 2, N.Y. Head, Technical Reference Section, U.S. Department of the Interior, Bureau of Reclamation, Denver Federal Center, Denver, Colo. Editor, Bibliography of Technical Reports, Office of Technical Services, U.S. De- partment of Commerce, Washington 25, D.C. Editor, Technical Data Digest, Armed Services Technical Information Agency, Document Service Center, U.B. Building, Dayton 2, Ohio Editor, Engineering Index, 29 West 39th Street, New York 18, N.Y. Editor, Aeronautical Engineering Review, 2 East 64th Street, New York 21, N.Y. Editor, Applied Mechanics Reviews, Midwest Research Institute, 4049 Pennsyl- vania Ave., Kansas City 2, Missouri Hydrodynamics Laboratories, Attn: Executive Committee, California Institute of Technology, Pasadena, Calif. New York University Institute for Mathematics and Mechanics, 45 Fourth Ave., New York 3, N.Y. Supervisor of Shipbuilding, USN, and Naval Inspector of Ordnance, 118 E. Howard Street, Quincy 69, Mass. Newport News Shipbuilding and Dry Dock Co., Newport News, Va., for distribution: 1 Senior Naval Architect 1 Supervisor, Hydraulic Laboratory Supervisor of Shipbuilding, USN, and Naval Inspector of Ordnance, New York Ship- building Corp., Camden, N.J. 1 Mr. J.W. 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Birkhoff, Department of Mathematics, Harvard University, Cambridge, Mass. Dr. J.V. Wehausen, Editor, Mathematical Reviews, American Mathematical Society, 80 Waterman St., Providence 6, R.I. Dr. David Gilbarg, Department of Mathematics, Indiana University, Bloomington, Ind. Prof. K.E. Schoenherr, Dean, School of Engineering, University of Notre Dame, Notre Dame, Ind. Mr. Maurice L. Anthony, Armour Research Foundation, 3422 S. Dearborn St., Chicago 16, Ill. Prof. N.M. Newmark, College of Engineering, University of Illinois, 111 Talbot Laboratory, Urbana, III. Dr. C.C. Lin, Department of Mathematics, Massachusetts Institute of Technology, Cambridge 39, Mass. Dr. Bruce G. Johnston, 301 W. Engineering Bldg., University of Michigan, Ann Arbor, Mich. Prof. Lydik S. Jacobsen, Department of Mechanical Engineering, Stanford Univer- sity, Calif. Mr. H. Ziebolz, Vice-President, Askania Regulator Co., 20 East Ontario Street, Chicago 11, Ill. Dr. Finn Jonassen, Technical Director, Ship Structural Committee, National Research Council, Washington 25, D.C. : Prof. M.L. Albertson, Head of Fluid Mechanics Research, Colorado A & M College, Fort Collins, Colo. 1 Prof. A. Yih Prof. M.A. Abkowitz, Massachusetts Institute of Technology, Cambridge 39, Mass. Mr. J.P. Breslin, Gibbs and Cox, 21 West Street, New York 6, N.Y. Prof. R.C. Binder, Department of Mechanical Engineering, Purdue University, Lafayette, Ind. Prof. N.W. Conner, North Carolina State College, Raleigh, N.C. Dr. F.H. Clauser, Chairman, Department of Aeronautics, Johns Hopkins Univer- sity, Baltimore 18, Md. VADM E.L. Cochrane, USN (Ret), Member, Panel on the Hydrodynamics of Sub- merged Bodies, Massachusetts Institute of Technology, Cambridge, Mass. Copies 55 CAPT W.S. Diehl, USN, Associate Member, Panel on the Hydrodynamics of Sub- merged Bodies, 4501 Lowell St., N.W., Washington, D.C. Mr. Hollinshead de Luce, Chairman, Hydrodynamics Committee, c/o Bethlehem Steel Co., Shipbuilding Division, Quincy 69, Mass. Dr. N.J. Hoff, Polytechnic Institute of Brooklyn, 99 Livingston St., Brooklyn 2, N.Y. Dr. Th. von Karman, 1051 South Marengo Street, Pasadena, Calif. RADM P.F. Lee, USN (Ret), Member, Panel on the Hydrodynamics of Submerged Bodies and Vice President, Gibbs and Cox, Inc., New York, N.Y. Dr. J.H. McMillen, National Science Foundation, Washington, D.C. Prof. J.W. Miles, University of California, Los Angeles, Calif. Dr. George C. Manning, Prof. Naval Architecture, Massachusetts Institute of Technology, Cambridge 39, Mass. RADM A.I. McKee, USN (Ket), Member, Panel on the Hydrodynamics of Submerged Bodies and Asst. General Manager, General Dynamics Corp., Electric Boat Division, Groton, Conn. Mr. J.B. Parkinson, Langley Aeronautical Laboratory, Langley Field, Va. Dr. W. Pell, Department of Mathematics, University of Kentucky, Louisville, Ky. Dr. M.S. Plesset, Hydrodynamics Laboratories, California Institute of Technology, Pasadena 4, Calif. Dr. J.M. Robertson, c/o Ordnance Research Laboratory, Pennsylvania State Col- lege, State College, Pa. Prof. A. Weinstein, Department of Mathematics, University of Maryland, College Park, Md. ; Prof. L.I. Schiff, Department of Physics, Stanford University, Calif. Dr. G.F. Wislicenus, Mechanical Engineering Department, Johns Hopkins Univer- sity, Baltimore 18, Md. Prof. W. Sears, Graduate School of Engineering, Cornell University, Ithaca, N.Y. Dr. Stanley Corrsin, Johns Hopkins University Department of Aeronautics, Balti- more 18, Maryland Dr. C.A. Truesdell, University of Indiana, Department of Mathematics, Blooming- ton, Ind. Prof. K.V. Laitone, University of California, Berkeley, Calif. Dr. L. Trilling, Massachusetts Institute of Technology, Cambridge 39, Mass. Prof. F.M. Lewis, Massachusetts Institute of Technology, Cambridge 39, Mass. Mr. M.A. Hall, University of Minnesota, Minneapolis 14, Minn. 56 Copies 1 Dr. A. Kantrowitz, Cornell University, Ithaca, N.Y. 1 Dr. R.T. Knapp, Hydrodynamic Laboratories, California Institute of Technology, Pasadena 4, Calif. 9 British Joint Services Mission (Navy Staff), P.O. Box 165, Benjamin Franklin Station, Washington, D.C. 3 Canadian Joint Staff, 1700 Massachusetts Ave., N.W., Washington, D.C. il Australian Scientific Liaison Office, 1800 K Street, N.W., Washington, D.C. il Director, Hydrodynamics Laboratory, National Kesearch Council, Ottawa, Canada 1 Prof. T.H. Havelock, 8 Westfield Drive, Gosforth, Newcastle-on-Tyne 3, England 1 Mr. C. Wigley, 6-9 Charterhouse Square, London EC-1, England 1 Dr. Georg Weinblum, Ingenieur Schule, Berliner Tor 21, Z 620, Hamburg 1, Germany 1 Capt. R. Brard, Directeur, Bassin d’Kssais des Carénes, 6 Boulevard Victor, Paris (15e), France il Dr. L. Malavard, Office National d’Etudes et de Recherches Aéronautique, 25 Avenue de la Division - LeClerc, Chatillon, Paris, France 1 Gen. Ing. U. Pugliese, Presidente, Istituto Nazionale per Studi ed Esperienze di Architettura Navale, Via de ta Vasca Navale 89, Rome, Italy 1 Sr. M . Acevedo y Campoamor, Director, Canal de Experienceas Hidrodinamicas, El Pardo, Madrid, Spain 1 Dr. J. Dieudonné, Directeur, Institut de Recherches de la Construction Navale, 1 Boulevard Haussmann, Paris (9e), France 1 Superintendent, Nederlandsh Scheepsbouwkundig Proefstation, Haagsteeg 2, Wageningen, The Netherlands 1 Prof. J.K. Lunde, Skipsmodelltanken, Tyholt, Trondheim, Norway 1 Prof. H. Nordstrom, Director, Statens Skeppsprovningsanstalt, Goteborg 24, Sweden 1 Director, British Shipbuilding Research Association, 5 Chesterfield Gardens, Curzon Street, London, W.1, England il Dr. J.F. Allan, Superintendent, Ship Division, National Physical Laboratory, Teddington, Middlesex, England 1 Dr. J. Okaba, The Research Institute for Applied Mechanics, Kyushu University, Hakozaki-machi, Fukuoka-shi, Japan 1 Prof. J.L. Synge, School of Theoretical Physics, National University of Ireland, Dublin, Ireland 1 Sir R.V. Southwell, 9 Lathburg Road, Oxford, England il Australian Council for Aeronautics, Box 4331 G.P.D., Melbourne, Australia Copies 57 Canadian National Research Establishment, Halifax, Canada Admiralty Research Laboratory, Teddington, Middlesex, England Armaments Research Establishment, Fort Halsted, Hants, England Prof. J. Ackeret, Institut fur Aerodynamik der Eidgenosssche, Technischn Hochschule, Zurich, Switzerland Prof. G.K. Batchelor, Trinity College, Cambridge University, Cambridge, England Dr. S. Goldstein, Haifa Institute of Technology, Haifa, Israel L. Escande, Ingenieur I.K.T., Professeur a la Faculte Des Sciences, Directeur de |’Kcole Nationale Superieure, d’Klectrotechnique et d’Hydraulique, 4 Boulevard Requit, Toulouse, France Prof. L. Howarth, Department of Mathematics, University of Bristol, Bristol, England Prof. Harold Jeffreys, St. John’s College, Cambridge, England Mr. W.P. Jones, National Physical Laboratory, Adrodynamics Division, London, England Prof. J. Kampe de Feriet, Faculte des Sciences, Universite de Lille, Lille (Nord) France Laboratorium voor Scheepsbouwkunde, Technical University, Nieuwe Laan 76, ri DELFT, The Netherlands, Attn: dr ir J. Balhan Head, Aerodynamics Division, National Physical Laboratory, Teddirgton, Middle- sex, England Head, Aerodynamics Department, Royal Aircraft Establishment, Farnborough, Hants, England Director, Aeronautical Research Institute of Sweden, Ranhammarsvagen 12, Ilsvunda, Sweden The University of Liverpool, Mathematical Institute, Department of Applied Mathe- matics, Grace Library, Liverpool, England Editor, Bulletin of the British Hydromechanics Research Association, Netteswell Road, Harlow, Essex, England NAVY-DPPO PRNC, WASH., D.C. ial Nie me eh wip ei 4 na) Senet igt ee “GW Herm ‘surmanp Alooy], - MOTJ pIn[y SeIpog punore Mozy Apeoysuy ‘Moya SJUSWO,| - Salpog [euoIzE}ONy ‘MOTT “q WeriM ‘surmunp Arooy J, - MoTj pry y SerIpog punoie MO] Apeoysug ‘mo, q SjUsWoW - SeIpog [euorje}0UT ‘MOT J pUuv S910] OY], “oull) YIM Sulsuvyo st yorym ureens *v ut (uorzR} -O1 SUIpNour) uoljow Apeoys-uou Aseaytqie uv oj yoolqns st Apoq eY} YOIYM UT eSB EY} 07 pepue;xe pue PeATIEped SI 4[NSeJ STYyy ‘ieded quoseid oy] uy *selzlie[nsuts ey] jo pooysoqysteu oy ur urea1}S PpeqINjSIpUN oy} JO JoJOBIBYD oY} pu’ SeTytIe[NZuIs ey} jo SyjBueNs oy} jo suorounj a[duits oq oj punoy e1om Aoyy, “Apog 043 JO sd8JMs OY} 0} JOTIEqUI SeTyTae[NSuts jo usysXs v Aq poyuesel -dei eq ues Apog oy) ueym ATpeSey Aq punoy o10Mm MOT] [eTUSIOd Apeoys Ayenique we ut peoed Apoq 8 uo yuewow pue 9010] ayy, GaIdISsSvVTONO *sjoi ‘*ss1j ‘yout *d yg fa “e967 ounp ‘uoyduryse A “SUIWUIND “| WeITTIM ‘AVAULS TVLINGLOd AUVULIGYV NV NI ONIAOW AGOd V NO ONILOV SINANOW GNV SHOUON GAL 082 “‘Idey ‘urseg [opow so[ABy, “4 praeg pue 9010} ey], *eul} YIM Surduvyo st yorym weens v8 ul (UOT) -01 SuIpnjour) uorjow Apeeys-uou Arentque ue 0} yoolqns st Apoq ey} YOIYM UT OSB OY] 07 pepuejXe pue PeATIepel SI 4[NSeI STYy ‘1eded queseid oy} uy ‘serjtre[nsurs ey} Jo pooysoqysteu oy ut WeeI]S peqinjstpuN ey} jo JeJOVIBYO eyy PUL SeI}TIe[NFuUTs oy} jo syjsue4s ey} Jo suoljoun] ejduits eq 0} punoj e1om Aoyy, *Apoq ey} JO O0BJMS OY} OF JOTIEJUI SoTITIB[NdUIS jo Weysks w Xq pequesei -dai eq uvo Apog ey} uoym AjpeSeq Aq punoj o10M MOT] [erUE}Od “Ti Apee}s Arentqie ue ut peoetd Apoq v uo yuowlou pus e010) oy J, GuIdISSVTONN : *“sjoi “*sS1j ‘pout ‘d jg ‘a a "eceT ounp ‘uojsutyse “SUIMUIND “H WeNTIM ‘NVAYLS TVILNALOd AUVULIGUV NV NI ONIAOW. AGOd V NO DNILOV SLINGNOW GNY SHOOT AHL “082 “dey ‘urseg Jopow sojABy “Mm praeg “G MeryyT iA ‘surwumD Arooy L, - #oTy PINT A Serpog punoie Moly Apeeysuy ‘MoT yy SqUSWOW - SeIpog [euorjzezOIy ‘MOTT “A Ue TIM ‘surmung Alooy], - MOT pnp SeIpog punole Mop Apeeysuy ‘Mop y SJUSWIOW - Selpog [euorjejONyT ‘MOTT pue 0010) OY, “Oully yZIM Sursuvyo st yoryM ueeys v ur (UOT}E] -01 SuIpnyoul) uorjow Apeoys-uou Ayexytqas us 07 yoelqns st poq 9} YOIYA UT eseO OY} 0} pepue;xXe pue peAlJeped SI 4[NSe1 SIyy ‘seded quosoid ot] ul “serjtaeynduts oy] Jo pooysoqysteu oyy ur WBeI}]S POqINySIpUN oY} JO JeJOVIBYO oY} puv SErzTIv[NFuIs ey Jo syjsueys ey} Jo suoTjouN] adults oq 0} punoy o1om Aayy, “Apog ey} JO eo¥jMs oY} 0} JOTIEqUT SeIqTIB{NBuIs jo weysks w kq powesol -doi eq uvo Apog ey} woym ATpeSeq Aq punoj o1om MoT] [eMUEIOd Apeo}s Areniqie ux ur peoeyd Apoq @ uo yuowou pue 60103 ayy, GaIMISSVTONA *sjea “‘sd1y -jour ‘d yc ‘A "eg6] ounp ‘uoyZuryseq “SUIMUIND “| WRIT ‘AVAALS TVLINGLOd AUVULIGUV NV NI DNIAOW AGO Y NO ONILOV SINGWOW GNV SHOOK FHL “081 “Jdoy ‘ulseg Jopow 10[ABT, “Mm pIAeq PUB S910} YT, “Sully YIM duIdusyo st yoy ulveNs v UI (UOTE -O1 SuIpnyout) uorjow Apeoys-uou Asexytqie ue 0} yooelqns st Apoq 64} YOIYM UT eSBd ey} OF pepuSszxe pus PeATIoped SI zJ[NSeI SIyy ‘reded queseid oy} uy ‘*selzlie[NSuts oyy jo pooysoqysteu oy ur UBe14S PEqINjSIpUN oY JO JeJOBIBYO oy} PUY SET]TIB[NSUIS oy} jo syjsue1s ey} Jo suoljounj ejduits eq 0} punoy e1om Aoyy, “Apog oy} JO edBJMs OY} 0} JOTEJUI SeTyTIe[NSuIs Jo weyshs ve Aq pequeser -dei oq uvo Apog ey} ueym AypeSe7y Aq punoy oom MoT] [etUS}Od Apeoeys Aivaqiqis ue ur peoeld Apoq @ uo yuewow pus 0010} eyy, GaI4IssvTONN *sjoi ‘*sd1j ‘pour ‘d yg ‘A “ec6T ounp ‘uoyduryseM “SUIWUIND “GY WeTTIM ‘NVAULS TVILNALOd AMVULIAAVY NV NI SNIAOW AGOE V NO ONILOV SLINGWOW CNV SHOOK AHL 082 “‘Idoy “urseg [opow Jo[AB], “Mm prAsq *quOWOW oY} 10J peitnbe st Apoq oy} Jo oovyjans *quoWOW oY} 10J peitnbea st Apoq ey} jo oowjins OY} JOAO UONBIEVUT UB yng ‘Apog oY} JO UOTJOW eYy JO puB UOT}IpuoD AsepuNog oy) surystqeyso 8Y} JOAO UOTyBISEWUL UB yng ‘Xpog oY} JO UOT}OW oY} JO pu’ UOT}IpUOD Azepunog ey} duryst{qeise UI pesN SsoNlieNsuls oy} JO SuOTyoUNy oTduits sv odJ0j OY} JO} UOAIS O18 SqUQUOdUIOD [Buoljtpps UI pesn solylis[Nsuls oy} Jo SuOTOUN o[duIIs sv oD10J EY} 10J UEATS oe syuoUOdUIOD [euoryIppe eseyT, “SsyWouodwoo [BuoIzIppe snyjd ,,jUeWOoW pus od10J A][eVdBT,, 04) oq 07 punoj ere JUSWIOUI esey], ‘sjueuodwoo jeuorjippe snqd ,,yUeuWOU pus edI0} A][B3e'T,, 949 Og 0} puno] ore yUoMIOW *quowloul oy} JoJ peatnbei st Apoq oy} jo eovjans *qUOWIOW OY} JOJ peitnbes st Apoq oy} jo oovjans ©Y} JOAO UOTBIsEqUT UB 4nq ‘Apoq oYy JO UOTJOW OY} Jo pue UOT}IpuoD Arepunog oy} surysttqeyso 84} JOAO UOTZVISEZUI UB yng ‘Xpog oY} Jo UOT}OW oY Jo pus uolqipuos Arepunog ey} Surysi{quyso UL pasn sorjzlie[NsuUIs oy} Jo suoTouny efdults sv 9010} oY} 10} UEATS ore s}UoUOdUIOD [euoljtpps UI pesn sorjzlaw[nsuls oy} Jo suorjouny o[dwits sv e010} oy) 10} UAT orev syuouoduloo [eBuolqIppe eseyt “s}ueuoduoo jeuorztppe snjd ,,jueWoW pus ed10j ATTedBT,, 04) Oq 07 puNoy O18 JUOUIOW ssey “s}ueuodwoo [wuortppe snyqd ,quewou pus eds0J A]]edeT,, ey} oq 07 puNo] ow QUOWOW “Y WelyTIM “surmanD ArooyL - Moy pmryq serpog punoie Mo, q Apeoeysuy ‘mop y SJUSUO - Selpog [euoryeOuy ‘MOTT “Gq WRIT] ‘suTMuND AlooyL, - MOTJ pny A SeIpog punoie MOT A Apeoysuy ‘Mop y S}JUSWOW - SOLIPOg [euoHyeiouy ‘MOT A pUv 9010] OY, “OUIt} YIIM Sutrsuvyo st yoryM urvons v Ur (UOT}E? -O1 SuIpNyour) uorjou Apeoys-uou Arvaytque ue 0} yoolqns si Apog OY} YOIYM UL eso oY} 07 popuoyxe pU PaATJOpal SI #[NSeI SIy} ‘zeded queseid ay} UL “Salzliv[NsUIs ey} Jo pooysiogysteu ou} ur WBOI}S peqinjstIpuN ay} jo JoyovIvYyO o4y pus SerTzIIv[NSurs oy) jo SYjBueIs oyy Jo suOTjoUN] aydwiIs eq 07 punoy oJom Aayy, “Apog 043 JO 9dBJMS oY} O} JOTIOJUI SOlzABNduIs jo wezshs v Aq poyuosel -dei eq uvo Apoq oy) ueym AT[esdey Aq punoy oJoM MOT] [VtyUOIOd Apeeys Arexjtqie ue ut pooryd Apoq @ uo queulow pus 990103 oY, GHINISSVIONN *sjoi “*sS1y our ‘dpc ‘A “eggT oune ‘uojsurysem *SUIMIUND “Y WeIT[IM ‘NVAULS TVILNALOd AUVULIGHV NV NI DNIAOW AGO V NO DNILOV SINGNOW GNY SHOYON ABL “082 “‘Idoy ‘urseg [opow so[Awy, “4 pravg pue e010) ey], ‘eulty yyIM Surduvyo st yorym ulvens v ur (uOTS] -01 Surpnyour) uorjou Apeoys-uou Aresytqie uv 07 yoolqns st Apoq 84} YOTYM UT OSvO OY} 0} popue}Xe pU¥ POATIJOpol SI y[NSEI SI} ‘jeded queseid oy} UL ‘SolzlIe[NSuIs yy Jo pooysoqysteu oyy ur WBEI]S PeqiINJsIpUN oY} JO JoyovIeYyO oYy PUY SOTJIIB[NFUIS oy Jo syjsues}s 64} Jo suotjounj a[duits eq 0} punoj e10m Aoyy, *Apog oy} JO edBjJMS OY} 0} JOMEJUT SoyriB[NZuts jo wosks B Aq poyuosos -dei eq uvo Apog ey} ueym ATpedeq Aq punoj o10M MoT] TeTUO}Od Apeeys Arentqiae ue ut pooetd Apog @ uo yuouOW pus 90103 oy J, GalaIssvTONA *sjor ‘*sstj ‘pout ‘d yg ‘a "ece] oung ‘uoyZuryse “SUIMUIND “| WeNTIM ‘NVAALS TVILNALOd AYVULIGUV NV NI ONIAOW AGOd VY NO ONILOV SINANOW GNV SHOUON AHL “082 “‘Idey “urseg [opon sopAL “MM pravq “@ WerqTiM ‘surmano Aro0y L - MOTy pny y Serpog punoie mopq Apeoysuy ‘Moy SqUSWOW - Selpog [Buotje}OIy SMO] LT “q Wely[IM ‘surmunD AlooyT, - MoTy pny Selpoq punois MO] \y Apeoysuy ‘Moy yy syUSUOP - SeIpog [euorjejo4y ‘MOT pus ed10] OY, “OWN YIM Surduvyo st yOrYA wWeens v ur (UOT) -01 SuIpnyour) uoow Apeeys-uou Arexyrque uv.oj yoolqns si Apoq OY} YOIYM UL eSed 94} 0} papuleyxe pue peAlJepoal SI 4[NSeI SIT} ‘ieded juoseid ay} UT “seljrie[nsurs oy} JO pooysoqusteu ayy ur Uva1jS PoqINySIpUN ey} JO Je}OBIVYO Oy] pus SeI}TIB[NFUIS ey} Jo Syjduexs ey} Jo suoTjouny afduits eq oj punoy o19Mm Aoyy, “Apoq ey4 JO ed¥8jMS olf} 0} JOLIEqUI SeTylIB[NBULS jo wWaysks vw kq paquesas -dei eq uvo Apog oy} ueym ATpesey Aq punoj o1oM MOT] [BTJUSEIOd Apeoys Arexqique we ut peortd Apoq 8 uo quowou pue e010; oyy, Ga4Iss¥TONO *sjoi “*sd1y ‘pour ‘d yg ‘A “E96T ounf ‘uojduTYsEM “SUIMWUND *Y WITT “NVEULS TVILNALOd AMVULIGUV NV NI ONIAOW AGO V NO ONILOV SLNGWOW GNY SHOU AHL "081 “Wdey ‘ulseg [epo 1ojAvy, 4 prasg pue e010} ey, “elit yQIM SuIduByo ST YOTYM WBENsS *B UT (UOT}E} -O1 Surpnyour) worjom Apeojs-uou Arexique uv 0} yooelqns st Apoq a4} YOIYA UL eSB OY} OF pepuezxe pU POATIOpoel SI 4[NSeI SI} ‘ieded yuosoid oy} UL ‘*seryle[NSuts oy} Jo pooysoqysteu oy} ur UES PEGINJSIPUN oY} JO JeJOBIBYO OY] PUB SOTTIB[NSUIS OY} JO syjsueijs ey} jo suotjounj e;dwis eq oj punoy oom Aoyy, *Apog ey} jO ed¥jNS EY} OF JOMIEVUI SerytiepNSuts jo weyshs v Aq poyuesol -dei eq uvo Apog oy} ueym Ajpeseq Aq punoj o1om MOT] [VIQUOJOd Apvoys Aveyniqie ue ut peostd Apoq v uo yUeWOW pus 9010} ayy, GaI4ISsVTONO *sjoi ‘sty pour ‘d yg ‘A “eS6T ounp ‘uoysurysem “SUTWUIND “GY WITT ‘AVAUXLS TVIINALOd AUVULIGAV NV NI DNIAOW AGO V NO ONILOV SINAWOW GNY SHOYON FHL “082 “dey -urseg [epow JopAvy, “M praeq *qUOWOW ey} 10} poestnboi st Apoq oy} jo oovjins *queuloW eYy 10j pestnbei st Apoq oy} jo oovjins 84} JOAO UOT}BIFOVUI UB ING ‘Apog oYy JO UOT}OW oY} JO pus UOT}IpUOD AyepuNoOg ey} Suryst{quyse 64} JOAO UOTyeIFE}UI UB nq ‘Apog ey} jo UOTJOUI oY} Jo pus UOT}IpuoD AsepuNog oy} SUIYSITqQByse UI posn SOlzIIB[NSUIS oy} JO SUOTJOUNy oTduITs SB 9910} OY} JOJ UEATS ole SyUoUOCdWOD [eUOT}Ipps UI pesN SoerzIIe[NSUIs oY} JO SUOT}OUNJ O[duUIs S¥ 9910} OY} OJ USAID O18 s}UoUOdUIOD [eUOT]IPps esoyy, ‘s}ueuodwoo yeuontppe snjd ,,quewouw pus ed10j A][BdBT,, oY eq 0} puNnoy ose yUOWOU eseyy, ‘s}ueuodwoo jeuorippe snjd ,,jUeuOW pus ed10j A[[ese]T,, OY} Og 07 PUNO] ore yUoUIOW *quaWOW ayy 10j pertnbo1i st Apoq oy} Jo eovjans *qUOWIOW oY} JOjJ poimnbe st Apog ey} jo oovjans 84} JOAO UOTBIFEJUT UB yNq ‘Apoq oYy JO UOTJOW oY} JO pu UOT}IpuoD AyepuNog oy} Surystqeyse 84} JOAO UOTYeISEqUI UB Ing ‘Apog ey} Jo UOTJOW BY) Jo pue UOTIpuoD ArepuNog oy} SuIYyst{qeyse UI peSN seljIB[NSUIs oy} JO SUOTJOUNJ O[duIIS S¥ 9d10Jj OYY I0J USATZ ore sjueuoduOD [eUOTIpps UI pesN selqtie[NSUls ay} Jo SuoTJOUNJ o[duIIs SB OdJ0J OY} 10} UOATS ov syuoUOdWOD [BUOTIppe eseyy, ‘syueuodwoo jeuortppe snqjd ,.yuewow pus ed10j A][BsB],, OY) eq 07 puUNO] ore quoUIOU eseyy, ‘Ssjueuodwod [euoljzIppe snjd ,,juewou pus 90103 ATTesvT,, oy) og 0} puNo] orw JUOWOU