Certain Problems of Two-Dimensional Physics. Ill 



9. In the case of a cylinder of dielectric material, we have 

 directly 



Vo= i {/(*) + f (r) } + § (JT (5) -F(r)} , 



V, = i {/<?) +/(t) } + §?{f(0)-F(t)}, 



where V , K are the external, Vi, K; the internal potential 

 and specific inductive capacity, and /and F are to be deter- 

 mined from the usual conditions of finiteness, &c. For the 

 boundary conditions are 



V — V ~K ° —JC 



when = t. 



If, however, the boundary be charged to surface-density <r 7 

 we must add to V* the term 



f£rV's + Y<>)M„. 



2tK 



Let us consider the simple case of an uncharged elliptic 

 cylinder of radius a and specific inductive capacity K, 

 surrounded by air, and in the presence of a line charge, E 

 per unit length, cutting the xy plane at the point whose 

 coordinates are «r , y . Let 



ri,= \(x — .v ) 2 +(y-yo) 2 }*, 



be the distance of the point x, y from the line charge. Then 

 in the neighbourhood of r x =zO we have 



V =-2Elogr 1 . 

 But, putting 



*o + l Vo = c cosn O + fo + «7o), 

 we have 



ri*=(x—x ) 2 +(y-y Q y 



= c 2 {cosh (X + £ +^)~cosh (A + f + ^o)} 2 



= c 2 {cosh (f— ft) — cos (t/ — 77 ) }{ cosh (2\ + f + £o) — cos(^ + ?? )}. 

 And therefore, in the neighbourhood of: the point ft, *7o> 

 logn = i log {cosh (f— ft) -cos (9— 9 )} 



— i log {2 sin i(6— Vo—igo) sin H*"— ^to + *to)} 

 = i ^g { [cosh (£- f ) -cos (77-970)] [cosh (f + ft)- cos (rj-Vo)]} 

 cosh (f-ft) -cos (77-770) 



ilog 



COsil (f+ ft) -COS (r/-77 ) ' 



