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 singu- 

 larities over the surface of the body.^° 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 general 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 ^(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 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 

 ^^. Then 



