666 

 where 



BELL SYSTEM TECHNICAL JOURNAL 



Co + C,l + C2/2 + . . . = y^22 ^ {k^n + y^x-V/^ + k{^ - /^2-). 



To the extent that most of the value of the integral comes from 

 that portion of the path of integration very close to r^ = ki' the 

 expansion in powers of / is valid. Replacing each /by — (d/dw) 

 we have then 



Now 



Q'--'M'''-^'£+''--^~ + 



dw R2 R2 R2 R2 



gikiRi 



Ro 



(9) 



7 



cos dz 

 Ro 



J ikiR2 



'rT 



where 7 = — iki cos d^, 



^3 gikiR2 



dw'^ Ri 



/ COS 9, o 1 «'^i sin^ d^ . \ 



(7' + 37-^^+2.3y-^^— +...J 



yikiRi 



~Rr' 



,ik\Ri 



~Rr' 



etc., and so 



Q, - 2(yCo + y'Ci + yG + + • • •) 



tikiRi 



R2 



+ 2 cos d,{ly'Co + 2yCi + 37'C2 + + •••) 



, ik2R2 



+ i/^i sin2 d,(l ■ 2y'C, + 2 • SyCs + 3 • 4y'Cs + + •••) 

 + higher order terms 



2^2^7 



>ikiR2 



ikiRi 



I d . iki sin- e^ d^ 

 + cos Oz- 7, :^o I , ■ n , 



+ higher order terms. (10) 



The second order term can be neglected if kiRo >> 1, say if 

 R > 20X. It will hereafter be supposed that this is the case. 



Multiplying numerator and denominator of the first order term by 

 i and canceling a ki we have finally 



