256 The Mathematical Electricians of the 



the same as Fourier's equation for the linear propagation of 

 heat : so that the known solutions of Fourier's theory may he 

 used in a new interpretation. If we substitute 

 v - /,2<V - i -j- \x 



y t/ > 



we obtain 



A, = (1 + -v/^l) (nCR)l J 



and therefore a typical elementary solution of the equation is 

 V = e -( nCR ^ x sin \2nt - (nCR)^x}. 



The form of this solution shows that if a regular harmonic 

 variation of potential is applied at one end of a cable, the phase 

 is propagated with a velocity which is proportional to the 

 square root of the frequency of the oscillations : since therefore 

 the different harmonics are propagated with different velocities, 

 it is evident that no definite " velocity of transmission " is to be 

 expected for ordinary signals. If a potential is suddenly applied 

 at one end of the cable, a certain time elapses before the current 

 at the other end attains a definite percentage of its maximum 

 value ; but it may easily be shown* that this retardation is 

 proportional to the square of the length of the cable, so that 

 the apparent velocity of propagation would be less, the greater 

 the length of cable used. 



The case of a telegraph line insulated in the air on poles is 

 different from that of a cable ; for here the capacity is small, 

 and it is necessary to take into account the inductance. If in 

 the general equation of telegraphy we write 



V = e nx ^~ l + P l , 

 we obtain the equation 



R (R* n* \i 



2l f (L* ~ CL) ; 



as the capacity is small, we may replace the quantity under the 

 radical by its second term : and thus we see that a typical 

 elementary solution of the equation is 



F= e i siu n{x - (CL)- 1 * t}; 



* This result, indeed, follows at once from the theory of dimensions. 



