174 Mr. 0. Heaviside on the 



comes into operation. The first term of (48) then increases 

 with t without limit, the rest ultimately vanishing. 



24. In general, the conditions imposed at the ends of a 

 cable, when there are no impressed electromotive forces, are 

 of the following form : — 



At #=Z, 



a . idv , n d 2 v , /KA , 



°= v+H ^ +n * p dj + < 5 °) 



Here m X) . . . , n u , . . , are constants, and' v is the potential at 

 any time. 



Supposing there are no intermediate conditions, there is a 

 single solution of the form 



t>=SAsin^-+&^e~^, .... (51) 



provided that the right-hand side of (51) can be made to 

 satisfy (49) and (50), and to equal /(#), an arbitrary function 

 of x when t = 0. 



It follows from (49) and (50) that 



tan&=- T" m3 f + 7n5 f"" — , • • (52) 

 1 — m 2 a + m±a — . . . 



t a n(a + ft)=- ^ a -^' + W «f— • ;■ . . (53) 



1— n 2 a ' + n±cf — . . . 



and from these tan a can be expressed similarly, say 



tana- lrh rf + hti tf„ mam > ■ ■ ■ W 



and the a's required are the + roots, real and imaginary, of 

 this equation. 

 Let 



1 C l • f a \ x 7 \ • ( c h x i\i 

 u = j I sm ( "J- + ^i ) sin ( -j- + 7; 2 ) tf #, 



where a 1; & 1? a 2 , b 2 are any two pairs of values of a and b. 

 Then, by integration, 



a i a z n/xa („ _l 7, \ „ Ao ^ _i_ 7, \ i tan fa + b 1 ) __ tan fa + b 2 ) 



«1 «2 



>s fa + &i) cos fa + h) \ 



a-idn , , rtanfti tank 



■r r COS b Y COS 6 2 -{ 



Substituting in (55) the values of tan(a + &) and tan b from 



-rcos^cos&jj \ L ^-^— ^-^ I. . (55) 



/ 



