578 Mr. C. E. Weatherburn : Problems in 



Having found <j>(s) from this equation, we determine rj by 

 (21) and then juu(t) from (16). The final potential is 

 expressible in terms of these as in the preceding section. 



The same method may be extended to the case in which 

 there are any number n of conductors whose individual 

 charges are given. Instead of the single equation (20) there 

 now appears a system of n equations 



f fji^dt + viB^Ei (t = l, 2, ...,n)* 



while <v(s) = W{s) + i Vi Yi'(s). 



From these equations the quantities rj are found as expres- 

 sions in which the functions 



( \ B.(ts)<f>{s)dsdt 



occur linearly, and the kernel of the integral equation cor- 

 responding to (22) becomes 



4~ &{*<*) + 2 M t - (*) ( K(ta) dt, 

 an s i— i | A 



where the functions Mi (s) are known functions of the points 



s of X. 



Part II. — Steady Flow of Electkicity. 



§ 6. General Problem^. — The general problem of the steady 

 flow of electricity under the exponential potential requires 

 the determination of a solution of the equation 



A 2 V-#V=0, (2) 



which is continuous along with its first derivative except in 

 the following particulars : — 



(A) There are certain surfaces, S, where the potential 

 experiences a sudden jump 2n. These " electromotive " 

 surfaces may, for instance, be the surfaces of separation of 

 two different metals such as copper and zinc. At S then 

 we may represent the behaviour of the potential by the 

 equation 



Y(s + )-Y{s-)=2 v . 



* f dt denotes integration extended over the surface of the tth 



conductor. 



f Cf. Weber- Riemann, he. cit. S. 437-438. Our solution of the 

 present problem follows Plemelj's treatment of the corresponding 

 problem for the ordinary potential. Cf. Plemelj, he. cit. S. 194-199. 



