574 Mr. G. E. Weatherburn : Problems 



in 



and under the influence of the same fixed external charges ; 

 but with the important difference that now there is in the 

 field one or more non-conducting surfaces X where, though 

 the potential is continuous, its normal derivative is discon- 

 tinuous* according to the formula ._ 



^-)- £+ £V) = 2 ^)' • • • (15) 



where e + and e~ are positive constants, s + and 5" points of 

 the inner and outer regions respectively indefinitely near the 

 point s of the surface S. We shall use the letters t and $ to 

 relate to points on the conducting surface ®, s and a to points 

 on the non-conducting surface 2. 



Our problem is to find the potential in the outer region. 

 This will be accomplished if we can determine the surface 

 distribution fx{t) over ©, the volume distribution 77 (p) through 

 the conductors, and a further distribution which will give 

 the required discontinuity at 2. As a discontinuous normal 

 derivative is characteristic of a simple stratum, we shall 

 endeavour to satisfy the requirements by introducing a 



simple stratum of density ^~(p(s) over 2, assuming then the 



field to be a homogeneous dielectric. 



As in the first problem r)(p) must be constant for each 

 conductor and equal to ^PP Z -. Let /(£) represent the normal 

 derivative at the surface (R) of the potential due to the volume 

 distribution 77 and the fixed external charges. The simple 

 stratum over % has at © a potential equal to 



J 



g{ta)<^>{a)dor 

 2* 



and a normal derivative 



li(tcr) <$)(&) da. 



j 



Then, since the potential within the conductor is constant, 

 the total normal derivative is zero, so that 



-/i(0+i &(«$)/*(■&)<# + /(*)+( h(ta)(j>(a)da = 0. 



* Cf. Weber-ftiemann, 'Die partiellen Differentialgleichuns?en der 

 mathematischen Physik,' Bd. i. S. 323-4 (5th ed.). Braunschweig, 1910. 

 Also Plemelj, Monatshefte fur Math, und Physik, Bd. xvdii. S. 188-194 

 (1907), where a similar problem is treated for the ordinary potential. 



