39 

 will also be Introduced. We then obtain from [95] the approximation 



U(t) = cos y(t) [96] 



Just as was done in the case of Munk's approximate doublet distribu- 

 tion we can improve upon this approximation in terms of an estimated longi- 

 tudinal virtual mass coefficient for the body. For this purpose we will first 

 derive a relation between this coefficient and the velocity distribution. 



Let T be the kinetic energy of the fluid when the body is moving 

 with unit velocity in the negative x-direction. Then 



2T = -pJJ^ dS = 27r/jJ P y^sinyds 



by [88]. Integrating by parts and substituting for d^/ds from [93] now gives 



2T = -ttpJV |f ds = no j" P U(x) y^xjds - A 



where A is the displacement of the body. But also, by definition, 2T = k A. 

 Hence 



A(l + k) = npfuix) y 2 (x)ds [97] 



•'ft 



This is the desired relation between k and U(x). 



Now suppose, as a generalization of [96], that an approximate solu- 

 tion of the integral equations [9^] is U(x) = C cos y . If this value is sub- 

 stituted into [97 1, we obtain C = 1 + k . Hence an improved first approxima- 

 tion to U(x) is 



U x (x) = (1 + k x ) cos y(x) [98] 



Equation [98] gives an exact solution for the prolate spheroid. 



SOLUTION OF INTEGRAL EQUATION BY ITERATION 



In order to solve [9^] by means of the iteration formula treated 

 in Reference 17, it would be necessary to work with the iterated kernel of 

 this Integral equation. Since this would entail considerable computational 

 labor it is proposed to try a similar iteration formula, but employing the 

 original kernel: 



U n+1 (t) = U n (t) + cosy(t)[l - f ^^ U n (x)ds] [99] 



