10 



since, for elongated bodies, a and b very nearly coincide with the body ends, 

 Hence 



C =-n-(i + k ) [25 



In practice an approximate value of k may be taken as that of the 

 prolate spheroid having the same length-diameter ratio as the given body. The 

 values of k for a prolate spheroid may be computed from the formula 24 



k = mkjJEZL^I . [26] 



1 x 2 VxF^i - xin (x + Vx^T) 



where X is the length-diameter ratio. Hence 



3/2 



c = r Jll '—. — = 27. 



x 2 /x^T - xin (x + Vx^T) 



The values of k versus X have also been tabulated by Lamb and graphed by 

 Munk. 25 



END POINTS OF A DISTRIBUTION 



A difficulty in determining the doublet distribution from Equation 

 [19] is that the limits of integration, a and b, are also unknown. In the 

 method of von Karman 10 the end points are arbitrarily chosen; Kaplan 13 takes 

 the end point of the distribution midway between the end of the body and the 

 center of curvature at that end. 



Kaplan based his choice on a consideration of the prolate spheroid. 

 Thus the equation of the spheroid of unit length and length-diameter ratio X, 

 extending from x = to x = 1 , is 



y^L-fx-x 2 ) [28] 



X 2 



The radius of curvature at x = is then — 5. The exact doublet distribution, 



2X 2 

 however, extends between the foci of the spheroid which are situated at dis- 

 tances 



X - Vx^Tl 



2X 



from the end points. Hence the error in Kaplan's assumption, 



