253, 254] ELECTROMAGNETIC WAVES. 535 



and obtaining the current from the equation 



/= - 



We thus arrive at the so-called electrostatic theory of propagation, 

 given in 1855 by Lord Kelvin in his paper " On the Theory of the 

 Electric Telegraph*," which established beyond question the feasi- 

 bility of an Atlantic cable. 



As we have seen in 250, harmonic disturbances are propagated 

 with a velocity proportional to the square root of the frequency. 

 There is therefore no definite velocity of propagation in a cable, 

 and there is liability of signals mixing with each other and losing 

 their character. We are however more concerned with the question 

 of how a single arbitrary short disturbance is propagated. If we 

 consider a cable with different constants, for which 



fiV ?PV' 

 K'R = 



Ttf* P)'/ 1 ' 2 ' 



by changing the variables by multiplying by constant factors we 

 may make one solution do for both. If we put 



T ' nT f kf 



w ~ Cvt//j (j (// 



we have 



K'RW 



b dt ~ a? dx 2 ' 

 so that V'(a>' t t)=V(x t t\ 



b_K'R 



a 2 " KR ' 

 that is 



/y/2 /v,2 



K f R^ = KR x -. 



t t 



Accordingly the time necessary to produce a given potential at a 

 distance x from the origin is proportional to KR multiplied by 

 the square of the distance. We quote Lord Kelvin's words : 



"We may be sure beforehand that the American telegraph 

 will succeed, with a battery sufficient to give a sensible current 

 at the remote end, when kept long enough in action; but the 

 time required for each deflection will be sixteen times as long as 

 would be with a wire a quarter of the length, such, for instance, 



* Proc. Boy. Soc., May 1855 ; Math, and Phys. Papers, Vol. n. p. 61. 



