536 THE ELECTROMAGNETIC FIELD. [PT. III. CH. XIII. 



as in the French submarine telegraph to Sardinia and Africa. 

 One very important result is, that by increasing the diameter of 

 the wire and of the gutta-percha covering in proportion to the 

 whole length, the distinctness of utterance will be kept constant ; 

 for R varies inversely as the square of the diameter, and K 

 (the electrostatical capacity of the unit of length) is unchanged 

 when the diameters of the wire and the covering are altered in 

 the same proportion." 



(The so-called " KR-L&w " has been applied to the theory of 

 telephony on long-distance land-lines, to which it is not at all 

 applicable, as has been shown by Heaviside. The use made of 

 this law by the chief electrician of the English telegraphs would 

 have prevented long distance telephony in England, even had there 

 been any long distances.) 



Guided by the conclusion announced above, we shall insert a 

 new variable, u = x/*Jt, and attempt to satisfy the equation (6) by 

 a function of u alone. 



We have 



dV = dVdu ss 19F^ 

 dt du dt 2 du V#* ' 



aF = aFaM = aFj^ 



dx du fix du * ' 



d# 2 ~ Vtf du* d t 

 so that our equation becomes 



KR dV d*V 



or 



KR 



= u. 



dV KR , dV n . u * 



lo %- = +C0nst ' = 



The integral of this equation is 



dV KR , 



lo %du- = ^ +C0n 



or, integrating from to u = xj*Jt, 



r^t , 



.6" du. 

 . 



