1G8 Mr. 0. Heaviside on the 



curve 2, fig. 3, from (31). In the last two £=$, and the even 

 o's are found by Tables. 



20. The two important solutions 



and 



77rr - ' 2,t2 * 

 v = Er 1 + 2Er 1 2cos-^-e "t", . .. . . (40) 



where, in (39), v is the potential at x at time t after the intro- 

 duction of E at x=0, both ends being to earth, and in (40) 

 v is the same when condensers of very small capacities r-^cl and 

 r 2 cl are interposed at the ends, there being no fault, may be 

 both deduced from the corresponding formula when the con- 

 densers are of finite capacity. Suppose initially the conden- 

 ser at x=0 to be charged to potential E, and the potential of 

 the line and the condenser at x = l to be zero, with no im- 

 pressed electromotive force in the system. Then at time t the 

 solution is 



« A . (am ,- \ ;.-!«. 



where 



and therefore 

 Also 



tan b = , tan (a + b) = — , 



r,a r„a 



, (r l -\-r 2 )a 



tana=- ^-~ (41) 



ErJcosfr , C n • ( ax , y\ 7 , a r 2 l cos (a + b) 



— - f-j x sm ( -=- + b )dx + x — ^ J - 



r } a Jo \ I / r 2 a 



C l . 2 (am \ cos 2 (a + b) 



Jo sm Tr + T" + ^ ^ 



A = 



,COS"0 



1 i-^ar 

 The result is 



. ax ax 



r l asm T -cos- J 



l+r,+* 



l+!t(l + f Wl i+ _* ) 



where the constant term arises from the zero root of (41). 

 Now, when r x =r 2 =0, the other + roots of (41) are ir, 2ir, 

 3-77, . . . ; and (42) then becomes the same as (40). But when 

 r 1 = r 2 = co , the roots are the same with the addition of a se- 

 cond zero root. In the general term of (42) make 



1 + cos a 



r 1 = r 2 = ■ = , 



a sm a 



