ELECTRICAL TRANSMISSION ALONG WIRES 21 



at the surface of each conductor.'' In the dielectric outside the 

 conductor, the potential $ satisfies Laplace's equation in two dimen- 

 sions; hence 



^ ,d_ 



dx^ dy 

 in the dielectric; and 



2 + ^2 )*=0 (24) 



A$= 0, {j= 1,2, ■■' n) (25) 



OTj 



at the surface of the jth conductor. Also 



EJdrj = - f^. ^drj = y a, (i = 1 , 2, • . . n) (26) 



the integration being carried around the surface of the jth conductor, 

 Qj being the charge per unit length on the jth conductor. 



The determination of $ from (24)-(26), when the geometry of the 

 conductors is specified, is a well-known two-dimensional potential 

 problem, for the solution of which very general methods are available. 

 The solution results in the form 



$ = 4>i{x, y)Qi -j- </)2(x, y)Q2 + • • • + ct>n{x, y)Qn. (27) 



That is, $ is a linear function of the conductor charges Qi ' - • Qn, 

 and the coefficients <^i • • • ^n are unique functions of the geometry 

 of the transmission system and are determinable by the usual methods 

 of two-dimensional potential theory. 



2. The continuity of il/„ and {l/n)Mr at the surfaces of the con- 

 ductors is analytically formulated by the equations 



±F- - ^ F 

 or OT 



dn iXc dn 



where fx is the permeability of the dielectric and fj-c that of the con- 

 ductor. These relations, it will be understood, hold at the surfaces 

 of all the conductors, i*' is a wave function which satisfies Laplace's 

 equation in two dimensions in the dielectric; thus 



^ In the following, t and n denote vectors tangential and normal to the conductor 

 surface respectively. 



