Electrostatics and the Steady Flow of Electricity. 575 

 The solution o£ this Fredholm equation is 



K0-/W+J Ht*)<i>(*)d* 



+ f H(*&) $/($) + ( h{$*)<l>(*)d*ld$, 

 which in virtue of (5a) reduces to 



/*(*) =/l(0 +f K(t*)<j>{a)d*, (16) 



where we have written 



/iW=/(0+f H(*)/ (*)<** (17) 



Turning our attention to the discontinuity at 2 we notice 

 that the potential due to t] and the fixed external charges has 

 a continuous normal derivative here equal to a known function 

 F(s). The simple stratum fi(t) over ® has a potential whose 

 normal derivative is 



I h(s$)fi($)d$. 

 . e 



Hence by the properties of simple strata we have for the 

 total potential V(/>) 



U|£<-->-iV>]-*<" 



1 I [S {s "> + S (s+) ] -J *(•»)*(•)«**+*(») +J ft(*»)j.(»)4». 



On substitution of the value of ji(S) from (16) the second 

 of these equations becomes 



i ff<'-)+ £ CO] -•(.) +J s [^ <>C«0>) dr. 



wiiere 



If the values of -, (fi ) and t— (s + ) derived from these are 



substituted in (15), we find that the required function <f>(s) 

 satisfies the equation 



.... (18) 



