Electrostatics and the Steady Flow of Electricity. 571 



The fixed external charges and the known volume distri- 

 bution create a potential whose normal derivative just inside 

 the surface of the conductors is a known function f (t) of the 

 boundary point t given by 



dq being an element of the space occupied by the fixed 



external charge of density ~—p(q), and dp an element of 



the volume oE the conductor. Since then the potential of 

 each conductor is constant, this normal derivative must be 

 equal and opposite to that due to the surface distribution 

 fi{t), so that by the properties of simple strata * 



-#i(0+J*(»)i<*)^=-/(0- ... (7) 



This is a Fredholm equation for determining fx(t). It has 

 a unique solution because all the characteristic numbers of 

 the kernel are numerically greater than anityf. This value 

 of fj.(t) makes the potential constant in each conductor, the 

 normal derivative being everywhere zero on the inner side 

 of the bounding surface J. The required solution of (7) is 



K0=/(0+JH +l (^)/(3)A ... (8) 



the suffix denoting the value to be assigned to \ in the 

 equations (3) and (4). The distribution of electricity is then 

 uniquely determined by (6) and (8). 



§ 2. Conductors to earth. Electrical Green's function. — 

 An interesting case of the foregoing is that in which all the 

 conductors are connected to earth and the fixed external 



charges reduce to a single charge — at the point a. In 

 this case P*=0, ^ = 0, and 7r 



In order to express that the surface electrification fi(t) is 

 that induced by the charge at « we may write it fi[ta). The 

 equation (8) then becomes 



fi(tu) = h{t*)+ jH +1 (tf)7A(Sa)tf$=H +1 (te), . (9) 



* Cf. Weatherburn, loc. cit. First paper, § 1, (6). 



t Loc. cit. § 3. 



% Loc. cit. §8,(14). 



