410 BELL SYSTEM TECHNICAL JOURNAL 



Substituting from (7) (11) in (l)-(3) we have, tlierefore, 



IdS f ff'P , d'Q , d'Q 



l-n-'Ky- \dz- dx'-dz ' dy'd 

 (13) II, = 0. 



. , Ids d r 



Ids d r^ Mrp) 



p -\- ^p' -\- 7- 

 Ids / c3-P d'Q 



e'^^'+^'dp 



2Tr\y-\dxdz dxdz" 



where 



(15) P= r J,{rpy^'''-+~''-£l 



Jo Vp- + 



7" 



and 



(16) 



= r Mrp)c^''^^^-J^ 

 Jo Vp- + y- 



= 7o[|7(/^ + c)]A'o[^7(^ - s)]. 



with i?- = r- + S-. 



The integral P is well known/ while Q is evaluated by a suitable 

 transformation of a Fourier integral.^ /o(s) = /o('-) and ivo(s) 

 = ^iriHo'-^^iiz) are the Bessel functions of the first and second kinds 

 for imaginary arguments as defined by G. N. Watson.^ In reducing 

 n^ to this form we use the differential equation ^ for Jq to obtain the 

 relation 



{:^^ -\- i^) Mrp) +pVo(/'p) = 0. 



The components of the electric force in the earth are obtained from 

 11 by the formula 



(17) E = graddivn - 7TI, 



■* See e.g. H. Bateman, "Electrical and Optical Wave-Motion," Cambridge, 1915, 

 page 72; or G. N. Watson, "Theory of Bessel Functions," Cambridge, 1922, page 416, 

 formula (2) of § 13.47, with // = and u = L 



"* G. A. Campbell, "The Practical Application of the P'ourier Integral," Bell 

 System Technical Journal, 7, 639-707; using pair 936 of Table 1, with a = 5, substitut- 

 ing X- for (g- — 4) in the integral of G, and generalizing the resulting integral to in- 

 clude complex quantities. 



" G. N. Watson, op. cit., pages 77, 78. 



' G. N. Watson, op. cit., page 19, formula (1") of § 2.13. 



