As is well known, Ranklne bodies are obtained by superposition of 

 these flows with a uniform stream so as to obtain a dividing streamline begin- 

 ning at a stagnation point. Without loss of generality we may suppose this 

 uniform stream to be of unit magnitude. This dividing streamline is the pro- 

 file of the Rankine body for which, by [7], the stream function is 







= -ly 2 + f |i(t)(-l +*=$) dt [14] 



The boundary condition, Equation [8a], then gives as the implicit equation for 

 the body 



|%(t)(-l + 2 Y t )dt = ly 2 [15: 



where now y 2 = f(x) and r 2 = (x-t) 2 + f(x). In order to obtain a closed body 

 the total strength of sources and sinks must be zero, i.e., 



In this case [15] becomes 



•6 



«(t) dt = 



V(tj*^ dt -ly* [15a] 



In general [15a] cannot be solved explicitly for f(x) when u(t) is 

 given. A practical procedure for obtaining f(x) for a given x is to evaluate 

 the integral numerically forvarious assumed values of f(x) and to determine 

 the value which satisfies [15a] by graphical means. 



When f(x) is prescribed [15a] may be considered as a Predholm inte- 

 gral equation of the first kind for determining the unknown function u(t) . 

 This equation will not be treated. Indeed it will be shown that, when con- 

 tinuous distributions are considered, it is a special case of the more general 

 equation for doublet distributions which will now be derived. 



DOUBLET DISTRIBUTIONS 



Let m(x) be the strength per unit length of a continuous distribu- 

 tion of doublets along the x-axis between the points a and b (see Figure 1). 

 The potential and stream functions may be taken as 



4> 



= I"* m(t) ^f dt [16; 



