Electrostatics and the Steady Flow of Electricity. 577 



Although 7] is unknown it is constant in virtue of (6). 

 The functions f(t), f\ (t), and -bt(s) may each be divided into 

 two parts, the one due to the external charges, and the other 

 due to r} and proportional to it ; so that we may write 



/i(0 = K(*) + <?Y(0)a. (m 



«x(*) = It'(*) + iyY'(«) i b. ' ' ' ' K J 



R, R', Y, and Y' being known functions of the points t 

 and s. The solution proceeds exactly as in the preceding case 

 up to and including the equation (18); but now the function 

 w (5) of the second member of (18) involves 7? which itself 

 depends on the unknown function </> (s). Given, however, 

 the total charge E on the conductor, we have 



JV(0^-N.D = E, (20) 



D being the volume of the conductor. On substitution of 

 the value of fi(t) from (16) this becomes 



v { j Y(t)dt+ D} + j R(<>8 4- §\H.(ts)cf>(s)ds dt*=% (20') 



so that 



E'-^K(t s )4>( s )d,dt 



where E' and B are constants depending only on the form 

 of the surface ©. Substituting this value of rj in (196), and 

 thence the value of -gt(s) in (18), we have for determining 

 <£(s) the integral equation 



(e- +e + )4>(s)-Ke--e + ) [ i j-G(s*)<j>{v)(l* ~^g^ (\ K{ts)<f>(s)dsdt~\ 



= 2tW-(6--6+){R'«+|'Y'(.)} (22) 



Our problem is then reduced to the solution of the Fredholm 

 equation 



(/>(*) +\J N(«r)<£(cr)<7cr =¥(*), . . . (22') 



2 



whose kernel N(scr) is given by 

 N(s<r)=— G(5er;- 



hile \= — 



e -t e 



^jH(to)A, 



and "^(5) is the second member of (22) divided by (e -f e + j. 

 Phil. Mag. S. 6. Vol. 30. No. 178. 0c*. 1915. 2 P 



