416 Dr Bromivich, Symbolical methods 



where in the last formula the relation* q^^ = has been used in 

 simplifying q^ coth q-J,. 



Hence ^- = ~ {I - A + U iqil)'} 



Vn hi 



^0 



a{^_ l~Uqjy 



U \ 1 - s + sqa -}- ^ {q^iy 

 where we write 1 for ajb in the small term containing (q^l)^ in the 

 denominator. 



Write for brevity y =^ 1 — s + sqa and then 



£ ^ ^ fl _ L- IM" + iJiiir 



Vo bl\ y y2 



Following Heaviside, we proceed to expand 1/y in powers of 

 sqa\{^ — s), and then retaining only terms in q, q^ (the terms in 

 q^, q-^ being identically zero), we find 



Vq hi 



— n + ^ nqa + n^q^a^ — | {qi^^inqa + -^-^ ' 



where, for brevity, we write w = s/(l — s). 



In the actual problem s is small (about J^) and so in the last 

 term of the formula we may put 



2n/(l ~s)^2n 



without sensible error, and then the term in l^ reduces to 



— ^ iqil)^ (Snqa) = — nPaq-^q. 



To interpret these formulae we use the results^ 



_ 1 3__i ^. 



^ ^y{^TKt)' ^ 2KtV{7TKt)' 



then q-f^q = q^ 



g _na 

 Vn bl 



Hence we find that 



a / n^a^ P \1 



{i-s)^{7TKt) V I^" "^ %^t)y 



which is a slight extension of Heaviside's formula, given for the 

 case when the thickness of the shell is negligible^ . 



* See § 4 (iv) below. 



t § 4 below (i) and (ii). 



1 Electromagnetic Theory, vol. 2, § 236, formula (39). The extension consists 

 merely in the addition of the last two terms in the bracket: that terms of this 

 type would be present could be foreseen from Heaviside's formula (27) in § 229 

 (for the corresponding plane problem). There is also the external factor ajb in the 

 present formula. 



