n 



v / x^T i 





2 A 4A 2 l6x 4 v 2A 



diminishes rapidly with increasing A. 



For the half -body generated by a constant doublet distribution (a 

 point source), Kaplan's assumption gives a poor approximation. Let a 2 be the 

 strength of the distribution. Then it can easily be shown from [19] that the 

 source is at a distance a from the end of the body (stagnation point), and 

 that, if the origin is chosen at the latter point, the equation of the half- 

 body is 



SMl - i (If + ^f (if + - fe»i 



k 



Hence the radius of curvature at the end is^a, so that Kaplan's assumption 



2 ^ 1 



for the start of the distribution gives -ra. This is in error by-?a. 



An approximate method for determining the end points of a distribu- 

 tion and its trends at the ends is given in Appendix 1 . The given profile is 

 assumed to extend from x = to x = 1 and to have the equation 



y 2 = a 1 x + a 2 x 2 + a 3 x 3 + ... [30] 



The doublet distribution is assumed to extend from x = a to x = b, so that 



< a « b < 1 , with a near and b near 1 , and to have the equation 



m(x) = c + c^x + c 2 x 2 + . . . [31 ] 



Only the trends of the distribution near the origin are discussed in Appendix 



1 . It is clear, however, that by means of a linear transformation the equa- 

 tion of the given profile can be expressed so that the end points of the body 

 exchange their roles. Hence the results in Appendix 1 can be applied to either 

 end of the body. 



The method of Appendix 1 consists essentially of expanding the inte- 

 gral in [19] about the origin and equating powers of t on the two sides of the 

 equation to obtain a series of equations in the unknowns a, c , c , c , ... . 



i i '012 



By applying the first four of these equations an approximate solution is ob- 

 tained in the form 



a = ^ [32] 



