the given meridian profile into a circle be determined. The velocity poten- 

 tial is then expressed as an infinite series whose terms are universal func- 

 tions involving the coefficients of the conformal transformation. Kaplan 16 

 has derived only the first three of these universal functions. 



Cummins of the David Taylor Model Basin is developing a method based 

 on a distribution of sources and sinks on the surface of the given body. This 

 method is also exact, but the labor involved in its application has not yet 

 been evaluated. 



Another exact method, based on a distribution of vorticity over the 

 surface of the body, is being developed by Dr. Vandry of the Admiralty Re- 

 search Laboratory, Teddington, England. The methods of both Cummins and 

 Vandry lead to Fredholm integral equations of the second kind, which can be 

 solved by iteration. 



The present writer has developed two new methods, an approximate one 

 in which an axial doublet distribution is assumed, and an exact one based on 

 a general application of Green's theorem of potential theory. Both methods 

 lead to Fredholm integral equations of the first kind for which a solution by 

 iteration has been discussed by the author. 17 Indeed, the consideration of 

 this iteration formula was initiated in an attempt to find more satisfactory 

 solutions of the integral equations of von Karman 10 and Weinig. 12 These new 

 methods will be presented, and, by application to a particular body, compared 

 with other methods from the point of view of accuracy and convenience of 

 application. 



FORMULATION OF THE PROBLEM 



We will consider the steady, irrotational, axially symmetric flow 

 of a perfect incompressible fluid about a body of revolution. Take the x-axis 

 as the axis of symmetry and let x, y be the coordinates in a meridian plane. 

 Denote the equation of the body profile by 



y 2 = f(x) [1] 



Since the flow is irrotational there exists a velocity potential <j> 

 which, for axisymmetric flows, depends only on the cylindrical coordinates 

 x, y and satisfies Laplace's equation in cylindrical coordinates 



