Prof. Thomson on the Electric Telegraph. 151 



and if we assume 



v = e kc'<p (16), 



•we have 



^^^1=^ 07). 



an equation, to the treatment of which the preceding investigations 

 are applicable." 



Extract from Letter to Prof. Stokes, dated Largs, Oct. 30, 1854. 

 "An application of the theory of the transmission of electricity 

 along a submarine telegraph-wire, shows how the question recently- 

 raised as to the practicability of sending distinct signals along such a 

 length as the 2000 or 3000 miles of wire that would be required for 

 America, may be answered. The general investigation will show 

 exactly how much the sharpness of the signals will be worn down*, 

 and will show what maximum strength of current through the ap- 

 paratus, in America, would be produced by a specified battery action 

 on the end in England, with wire of given dimensions, &c. 

 " The following form of solution of the general equation 



, dv d^v , 

 kc— — —- — hv, 

 dt dx"^ 



which is the first given by Fourier, enables us to compare the times 

 until a given strength of current shall be obtained, with different 

 dimensions, &c. of wire ; — 



v = e~i'e . 2A: sin f v-r ) ■ c '":('. 



in)- 



If / denote the length of the wire, and V the potential at the end 

 communicating with the battery, the final distribution of potential 

 in the wire will be expressed by the equation 



t;=V 



which, when A=0, becomes reduced to 



corresponding to the case of perfect insulation. The final maximum 

 strength of current at the remote end is expressed by 



V 



or, when A=0, y= rr • 



Hence if we determine A, so that 



* See the diagram of curves given at p. 156. 



