40 



where r 2 = (x - t) 2 + y^x) and x = x(s). 



Here also it is convenient to express the iterations in terms of 

 error functions E (t) defined by 



c p U (x) y 2 (x) 

 i (t) = 1 - -^ ds [100] 



n J n ?T» 3 I 



2r° 



or, from [99], 



Hence 



Also, from [99], 



E n (t) cosy(t) = U n+1 (t) - U n (t) [101] 



n 

 U n+1 (t) = Ujt) + cos y(t) £ E ± [t) [102] 



ri E (x) y^x) 



where x Q , x are the nose and tail abscissae. Thus, to obtain U +1 (t), we 



first obtain E (t) from U (t) in [100], then E , E . ... E from [103], and 

 i i 23 n l-'j. 



finally U n+1 (t) from [102]. 



NUMERICAL EVALUATION OP INTEGRALS 



In applying Equations [100] and [103] it will frequently be neces- 

 sary to evaluate integrals of the form 



pE(x) y^x) dXj where r2 = (t _ x)2 + ^ (x) 

 \ r 3 



This form, however, is unsuited for numerical quadrature for elongated bodies, 

 since y 2 (x) peaks sharply in the neighborhood of x = t . Here, as in the case 

 of the integrals for the doublet distribution, two procedures are available 

 for avoiding this difficulty. The first employs the polar transformation [70], 

 involves several graphical operations, but in general transforms the integrand 

 into a slowly varying function so that the integral can be evaluated by a 

 quadrature formula using relatively few ordinates. The second retains the 

 original variables and eliminates the peak by subtracting from the integrand 

 an integrable function which behaves very much like the original integrand in 

 the neighborhood of the peak. The numerical evaluation of the resulting inte- 

 gral on the second method requires a quadrature formula with more ordinates 



