peak sharply in the neighborhood of t = x. In the range of the peak, in which 

 the value of the integral is principally determined, m(t) will vary little 

 from m(x). Also, only a small error will be introduced by replacing the lim- 

 its of integration by - °° and + <». Hence, as a first approximation to a solu- 

 tion of [19], try 



li(x )j -= T [21 



We obtain 



m^x) ^y 2 [22] 



a distribution proportional to the section-area curve of the body. This ap- 

 proximation was Independently derived by Weinig 12 who employed it as the first 

 step in a divergent iteration procedure. It has also been rediscovered by 

 Young and Owen 15 and Laitone 19 who have shown the accuracy of the approxima- 

 tion for elongated bodies by several examples. 



It is apparent from its derivation that [22] also gives the asymptot- 

 ic radius of the half -body generated by a constant axial dipole distribution 

 extending from a point on the axis to infinity. It is readily seen that this 

 distribution is equivalent to a point source at the initial point. 



As a refinement to Munk's formula, Weinblum 20 has used the approxi- 

 mation 



m x (x) =, Cy 2 [23] 



where C is a factor obtained by comparison of the distributions and section- 

 area curves of several bodies. Weinblum' s factor bears an interesting rela- 

 tion to the virtual mass of the body. This is seen by considering the expres- 

 sion for the virtual mass k A in terms of the mass of the displaced fluid A 

 and the totality of the doublets, f mdx, 



21 ,22,23 



6 



■ :.. 



k,4 = kirp 



f mdx - A [24] 



•'a 



where ^ is designated the longitudinal virtual mass coefficient, and p is the 

 density of the fluid. But, from [23], 



^np[ n^dx = 4pcf 7ry 2 dx = 4C4 



