37 



which, by comparison with [4], is seen to be the equation satisfied by the 

 Stokes stream function. Conversely, if Q is a function satisfying [4], it can 

 readily be verified that the functions u and \jj defined by [92] are correspond- 

 ing axisymmetric potential and stream functions, i.e., that they satisfy Equa- 

 tions [3]. Written in terms of Q, [90] now becomes 



•'O 'II 



If we choose for £2 the stream function of a source of unit strength 

 situated at an arbitrary point of the axis of symmetry within the body, we 

 have, from [10], 



Q= -1 +^, r = [(x-t) 2 + y 2 ] 1 ' [93] 



Then 



dx r 3 



and, since y vanishes at both limits, 



Hence [93] becomes 



r 



U(x) y 2 (x) 

 2r 3 



ds =1 [94; 



It is seen that [94] is an integral equation of the first kind in which the 

 unknown function is U(x) and the kernel is y 2 / '2r 3 . 



In contrast with the integral equations for source-sink or doublet 

 distributions which can be used to obtain the potential flow about bodies of 

 revolution, the integral equation [94] has two important advantages. The 

 first is that a solution exists, a desirable condition which is not in general 

 the case when a solution is attempted in terms of axial source-sink or doublet 

 distributions. The second advantage is that [94] is expressed directly in 

 terms of the velocity along the body so that, when U is determined, the pres- 

 sure distribution along the body is immediately given by Bernoulli's equation 

 [69]. In the case of the aforementioned distributions, on the other hand, it 

 would first be necessary to evaluate additional integrals, to obtain the ve- 

 locity along the body, before the pressures could be computed. 



