Secondly, the second term involving the integral of <t> is evaluated. The 

 asymptotic behavior of the integral over 8 = (O.vir) and at large distance 

 r^ 00 can be found by the method of stationary phase. The integral, after sub- 

 stituting <J> from Equations 9, 10, and 12, can be written as 



VTT 7T-a 



/n6 JQ f \ ikr cos(8-a) , ikr cos(8+a)~| n 6 



^ cos _ d6 = / |_ e + e J cos — 



■tr+a 



/ikr cos(Q-a) n8 , Q 

 e cos — d° 



(22) 



In the integrals, there are three points of stationary phase at 6 = a and 

 8 = t\ ± a . If the same argument as that of Stoker (1957) is followed, of the 

 three contributions only the first one 8 = a furnishes a nonvanishing con- 

 tribution for r^°° when the operator ^(S/Sr + ik) is applied to it. The 

 physical significance of this statement is that only the incident wave is 

 effective in determining the cosine coefficients of the solution. Therefore, 



vir 



i- J~(^ -l. -i\ f a n6 jo t/ttt na i(kr+ir/4) ,„_. 



l^m ^[j^ + lkl / <)> o cos — d8 ~ 2/2Trk cos — e (23) 



Substituting Equations 21 and 23 into Equation 19, we obtain the unknown co- 

 efficients a : 



0aT na iniT/2v 



a = 2ir cos — e (24) 



15. Since the solution <f> in the cosine series expression is 



Mr, 8) = L. HVf0) + L. ^ ^ (r>n) cos ]1° (25) 



n=l 



10 



