150 Royal Society : — 



and if a be infinitely small, this becomes 



^^Qa z£^ (13), 



277* i^ 



•wbich with positive values of z, expresses obviously the effect of 

 communicating the point O with the positive pole for an infinitely 

 short time, and then instantly with the ground. 



"The strength of the current at any point of the wire, being equal to 



— , as shown above, in equation (2), will vary proportionally 



a: ' dx 



to — or to — . The time of the maximum electrodvnamic effect 



dx dz 



of impulses such as those expressed by (11) or (13) will be found 



by determining t, in each case, to make — a maximum. Thus 



we find 



as the time at which the maximum electrodynamic effect of connect- 

 ing the battery for an instant at O, and then leaving this point in- 

 sulated, is experienced at a distance .v. 



" In these cases there is no regular ' velocity of transmission.' 

 But, on the other hand, if the potential at O be made to vary regu- 

 larly according to the simple harmonic law (sin 2nt), the phases are 



propagated regularly at the rate 2* /— , as is shown by the well- 

 known solution 



v = e~""' sm(2nt—zn^) (14). 



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 i,t 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 Farada)''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 ^„ ^.^ 



Ac — = hv (15), 



dt dx'' ^ ' 



* We may infer that the retardations of signals are proportional to the squares 

 of the distances, and not 'to the distances simply; and hence different observers, 

 believing they have found a "velocity of electric propagation," may well have 

 obtained widely discrepant results ; and the apparent velocity would, ctsteris pari- 

 bus, be the less, the greater the length of wire used in the observation. 



