387 



The effects of pulses at one end, when the other is in connexion 

 with the ground, and the length finite, will be most conveniently 

 investigated by considering a wire of double length, with equal positive 

 and negative agencies applied at its two extremities. The synthe- 

 tical method founded on the use of the solution (11) appears per- 

 fectly adapted for answering all the practical questions that can be 

 proposed. 



" To take into account the effect of imperfect insulation (which 

 appears to have been very sensible in Faraday's experiments), we 

 may assume the gutta-percha to be uniform, and the flow of electri- 

 city across it to be proportional to the difference of potential at its 

 outer and inner surfaces. The equation of electrical excitation will 



then become 



, dv d*v , ,,.,. 



kc = hv (15), 



dt dx* 



and if we assume 



_A 

 v=e *<fy (16), 



we have 



*<%=% (> 



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 



* See the diagram of curves given below. 

 VOL. Vll. 2 P 



