POTENTIAL ANALOGUE METHOD OF NETWORK SYNTHESIS 335 



and for the interior potential a suitable power series expansion is 

 P Fiip) = oo+T.amP'^. (37) 



m=-l 



Outside the circle we have \ p \ > Pn , so that the logarithmic terms may- 

 be expanded in convergent series of pn/p, 



Feip) = constant — 2^ ^n log ^ — £ ^n log ( 1 — ^) 



n n \ P / 



= .o-Mog.-i:.„(-^--|--). 



Hence a suitable power series expansion for the exterior potential is 



Feip) ^ bo-bo\ogp+ Z M"". (38) 



The constant bo represents the total charge on the circle. If there is no net 

 charge the logarithmic term vanishes and Feip) is analytic outside C. It 

 will vanish at infinity if we also have bo = 0, but for the moment we shall 

 retain both constants, and apply the boundary conditions on C to determine 

 the unknown constants bm from the known constants dm . 

 On the circle of radius coo we have 



p = cooe^'', (39) 



so that just inside C the interior potential is 



Fm = ao+ Za^o)?^*"*", (40) 



w»=l 



while just outside C the exterior potential is 



FM =bo-bo log (a,o^*') + E bmc^o'^e-'"'', (41) 



In our apphcations the constants a and b are real, hence we may separate 

 the real and imaginary parts of (40) and (41), and find 



Viii}) = oo + Z (^moy^ cos w^, ^ii^) = Z) ^m(^o slu Mt}, (42) 



Vei^) = &0 — ^0 log I COo I + Z bmOio"" COS M^, 



^M = -&ot> - Z bmo^o"" sin mi}. (43) 



The condition that V must be continuous across C determines the b's: 



bo — bo log I coo I = flo , bm = o)Tam , m> 0. (44) 



