62 Mr. W. A. Price on Alternating 



the current at A is 



E a 



—7= .- { 1 — 2e- 2aL cos 2aL H- ?- 4aL } § cos cat. 



V2 P 1 ' 



Changing the time from which t is reckoned, so that cat 

 becomes oat -f <f>, where 



1 — g- 2aL (cos 2aL + sin 2aL) 

 tan ^ ~ 1 - e -' JaL {cos 2*L -sin 2«L)' 

 and writing 



i.-{l-26;- 2aL cos2aL+e- 4aL }i as A, and L 7 as F, 



V2 P 

 the current at A becomes A cos cat ; 

 the potential at A becomes 



A T (sinh 2«L + sin2aL)sinw£ + (sinh2aL--sin2aL)cosrt)£ 

 ^p a cosh 2aL— cos 2aL 



and the potential at becomes 



2 A T (cos aLsinhaL+sinaL cosh aL) sin cb^+ (cos aLsiuhaL — sin aL cosh aL) cos u,t 



~~ f^ 01 cosh2aL — cos2aL • 



col 



The form and notation of these expressions are those given 

 in ' Alternating Currents of Electricity/ T. H. Blakesley, 

 London, 1889. 



Consider the circular electrical conductor AB (fig. 2), having 

 a total resistance 2tjt and a total 

 capacity 27rc: the character- 

 istic equation being 



<Pu_ du 



de 2 ~ rc 'dt' 



At a time t = 0, when the con- 

 ductor is uncharged, a charge 



27rS is introduced at A. The charge at any point P at a 

 subsequent time t is 



00 1 



s = S X cos We'^' 1 , where a= — : 



re 



sd& being the charge at P, where AOP = #, of a small arc 

 subtending dd at the centre. 



If a series of charges, 27rS cos cot . dr, is introduced at A 

 at intervals of time dr beginning when t = 0, s, the charge 

 at P at a time t, 



r*. £ -n f/-" 2 cos cat + ca sin cat fjui 2 ,„,") 



= S t COS 10 ] £ 2 , 2 . 4 ■ 2 , 2 * e ~* i • 



