Eggers 



we have 



-co .00 II 



1/r- l/rj - 7o/(277) e ° 2 sin (VyQZ) sin (VygO cos [U7o(Y - tj)] /U dUdV 



(AlO) 

 and 



.00 CO 11 



(1/r- 1/ri)^ -7^/(2^) ^ °' 2 sin (V7oZ) sin (V7oO 



IJ. 



X cos [U7o(Y-'^)] dUdV sign(X- ^) . (All) 



A general law in the theory of Fourier transforms (see Ref. 11 of the preceding 

 list) — essentially known as Poisson's summation rule — states that if the func- 

 tion F(y) has a representation 



F(y) = G(U) e^^y dU , (A12) 



then 



OP 



F (S,y) = 2_, ^^y+ ^^) • 

 provided this series converges, has a representation 



CO 



F*(S,y) = 2] ^("v) e'^^^AU (A13) 



with Au - l/s and u^ = z^Au. With S = tA, we therefore have the representation 



CO , - 



J2 |[(X- <f)2+(Y + 1^1- 7^)2 + (Z+O']''^'- [(X-f)2+(Y+vT-77)2+(Z- 0']''^'[ 

 2] e""^o'^'^' sin (V7oZ) sin (V7o0/U dV cos {V^y^i^ - v)) ^^^^^ 



,-■-00 v = -<» 



with u^ = 27rj>/(7o , T) and u = \A^^~i^lJ7^. But the terms under summation are 

 equivalent to corresponding terms in the series for G given by Eq. (Al), and it 

 can be seen that after subtraction of these terms the integrals for the coeffi- 

 cients g^°^^^ in Eq. (Al) converge even in the case x= -^ , and z= ^. 



The argument for the function c^^ is analogous. 



* * * 



672 



