and 



f.y-f^dt [17: 



Jo r 



The stream function for a Rankine flow now becomes 



»-4» , + »'f^« 



Hence the boundary condition, Equation [8a], gives 



£sit) dt = ^ [i 9 ] 



Here again Equation [19] may be considered as an implicit equation for the 

 Rankine body when m(t) is given, or as a Fredholm integral equation of the 

 first kind when the body profile y 2 = f(x) is prescribed. 



In order to show the relation between the source and doublet distri- 

 butions in Equations [15a] and [19]. integrate by parts in [19]- We have 



| m (t)^dt = m(t) — | a +J a 3t — dt 

 Hence [19] may be written as 



»">¥[ ♦ ft ^ dt =T^ 



20 



The interpretation of Equation [20] is that a doublet distribution of strength 

 m is equivalent to a source-sink distribution of strength dm/dt together with 

 point sources of strength m(a) and -m(b) at the end points. Hence source-sink 

 distributions are completely equivalent only tc those doublet distributions 

 which vanish at the end points. This justifies the remark in the previous 

 section that the integral equation for the doublet distributions is more gen- 

 eral than that for the source-sink distributions. 



MUNK'S APPROXIMATE DISTRIBUTION 



Munk 18 has given an approximate solution of Equation [19] for elon- 

 gated bodies. His formula may be derived as follows. At a great distance 

 from the ends of a very elongated body, the integrand of [19]. m(t)/r 3 , will 



