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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 undisturbed 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 changing with time. The force and moment are found to be the "Lagally force and mom.ent" 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 surface 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 potential flow of an ideal fluid has received considerable attention, the problem being of considerable 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. ToUmien developed a solution for the force and moment in terms of the "Kelvin impulses" and extended the discussion 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 ToUmien'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 nonuniform streamSjbut 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. References are listed on page 48. 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 witti a som.ewhat less mathematical approach than Lagally and presented the results in a very convenient form.^ Mohr discussed distributions of singularities over the surface of the body.^° Brard has recently attempted to extend Lagally's method 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 difficulties 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 obtain 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 general singularities, and the analysis is extended to the case of non-steady streams and non-steady motions of the body (rotation 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 additional force is stated in simple form in terms of the strengths of the singularities and the motion 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 ^(x, y,z, t) 2. If the body were not present, the stream would have a velocity potential oi, 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 system 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 ^^. Then 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 position vector of any point of space with respect to the stationary system is designated by r and with respect to the moving system by r^. , The position vector of the moving origin is r^, and its velocity is v^. The following relations hold; y-yo = y,, z ~ z ^^ z. [2] 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 absolute velocity by 'Im = 'I ^o where q^ is the relative velocity and q is the absolute velocity. Since v^ is a function of time only, it satisfies the identity

The forces and moments acting on a body moving in an arbitrary potential stream / by William E. Cummins.