19 



NUMERICAL EVALUATION OP INTEGRALS 



In order to perform the iterations in Equations [56] and [58] and to 

 compute the velocity distribution it will frequently be necessary to evaluate 

 integrals of the form 



fni(t) dt and r 6 m(l) dt 



Ja r> 3 Jo T> 5 



where 



r 2 = (x-t) 2 + f(x] 



Because in this form these integrals peak sharply in the neighborhood of t = x, 

 especially when the body is elongated, they are consequently unsuited for nu- 

 merical evaluation. 



A more suitable form can be obtained by means of the following trans- 

 formation. Let (x, y) be the coordinates of a point on the body, t the ab- 

 scissa of a point on the axis, $ the angle between a line joining these two 

 points and the x-axis; see Figure 1 . Then 



x - t = y(x) cot [70] 



We may now transform the integrals so that 9 becomes the variable of integra- 

 tion. Then 



and 



where 



f JL m (t)dt = [ m(t) sinfldtf [71 



Ja r 3 J° 



rb ,,4 r0 



*- m(t) = m(t) sin 3 6 dO [72] 



Ja r 5 J» 



a = arctan ^, = arctan -^ [73] 



An alternate procedure, which eliminates the peak without a trans- 

 formation of variables, is the following. We have 



f'-Z^ m(t)dt = P-Zf [m(t)-m(x)l dt + m(x) f* t- dt 

 Jo r 3 Jo r 3 L J Jo r 3 



