An Interesting attempt to solve the direct problem was made by 

 Weinig 12 in 1928. He also formulated the problem in terms of an integral equa- 

 tion for an axial doublet distribution which would generate the given body, 

 and employed an iteration formula to obtain successive approximations. Since 

 the successive approximations diverged, the recommended procedure was to extra- 

 polate one step backwards to obtain a solution. 



In 1935 an entirely different approach, in which a solution for the 

 velocity potential was assumed in the form of an infinite linear sum of orthog- 

 onal functions, was made by Kaplan 13 and independently by Smith. 14 The coeffi- 

 cients of this series are given as the solution of a set of linear equations, 

 infinite in number. In practice a finite number of these equations is solved 

 for a finite number of coefficients, and Kaplan has shown that the approximate 

 solution thus obtained is that due to an axial source-sink distribution which 

 is also determined. A simplification of Kaplan's method by means of addition- 

 al approximations was proposed by Young and Owen 15 in 19^3- 



It appears to be generally agreed, by those who have tried them, 

 that the aforementioned methods are both laborious and approximate. Thus, ac- 

 cording to Young and Owen: 15 



''In every case, however, the methods proposed are laborious 

 to apply, and the labour and heaviness of the computations 

 increase rapidly with the rigour and accuracy of the proc- 

 ess. Inevitably, a compromise is necessary between the 

 accuracy aimed at and the difficulties of computation. All 

 the methods reduce, ultimately, to finding in one way or 

 another the equivalent sink-source distribution, and it is 

 this part of the process which in general involves the 

 heaviest computing." 



Furthermore, a fundamental objection is that only a special class of bodies of 

 revolution can be represented by a distribution of sources and sinks on the 

 axis of symmetry. According to von Karman: 10 



"This (representability by an axial source-sink distribu- 

 tion) is possible only in the exceptional case when the 

 analytical continuation of the potential function, free 

 from singularities in the space outside the body, can be 

 extended to the axis of symmetry without encountering 

 singular spots." 



The dissatisfaction with these methods is reflected by the continuing attempts 

 to devise other procedures. 



A new method published by Kaplan 16 in 19^3 is free of the assumption 

 of axial singularities and appears to be exact in the sense that the solution 

 can be made as accurate as desired, but the labor required for the same ac- 

 curacy appears to be much greater than by other methods. The application of 

 the method requires that first the conformal transformation which transforms 



