Equation 73 may be written in the form 



dG 



1+ eG = 



dy \a)'^ - f^ dy 



We choose the origin of the coordinate system so that y = v„ is located 

 at f=0 (equator) and y = y^ atf=6j. With the transformations 



Y = 



1 > 

 7 J 



1 r^ ,, ,, 1/2 



G, e=-/ (<o^-y^^ dy, 



1 /I 

 J = - f (a 



' ^-tI 



, 1/2 



(76) 



(77) 



for the area cj ^ f v/e get from (75) the Schrodinger equation 



with 

 and 



d^Y 



+ [K^ -1(^)]Y = 



de 



K'^ = i' 



1/4 d- 



IP^ 



fco2-/-2)' 



ff"+f'^ 



(ify 



i^'^-f^r 2 (,;^-{^)- 



(78) 



(79) 



(80) 



Because of the smallness of /' and {" ,T(0 is negligible in regard to K'^ 

 for all areas outside of f , where oj = f{0 ■ Therefore, we suppose T(f) = 

 for these regions and get 



Y(^) = Ygexp (ikO, w > f (81) 



G (y) = Gq ra;2 - /2^ exp 



^v^/^/- 



[co^ - /-^ (v) ] dy 



(82) 



In order to find the influence of the singularity at cf , we consider (78) to be 

 an inhomogenous equation 



d2_Y 

 d^ 



+ K^Y = T(0 Y 



(83) 



with the solution 



Y'(0) 

 Y(f)=y(0)cos fe.f + — — sin/ef , 



k k 



+ — J sin k (^ - ^*) T(^*) Y(^*) d^* (84) 



T has the character of a S - function: 



27 



