152 Royal Society : — 



SA, sin {^'Tfj-j =-V ^;vA_^-,vA ^^en x>0 and x<l, 



the equation 



■will express the actual condition of the wire at any time t after 

 one end is put in connexion with the battery, the other being 

 kept in connexion with the ground. 



" We may infer that the time required to reach a stated fraction 

 of the maximum strength of current at the remote end will be 

 proportional to kcP. We maybe SM?-e beforehand that the American 

 telegraph will succeed, with a battery sufficient to give a sensible 

 current at the remote end, when kept long enough in action ; but 

 the time required for each deflection will be sixteen times as long as 

 it would be with a wire a quarter of the length, such, for instance, 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 n varies 

 inversely as the square of the diacieter, and c (the electro-statical 

 capacity of the unit of length) is unchanged when the diameters of 

 the wire and the covering are altered in the same proportion. 



" Hence when the French submarine telegraph is fairly tested, we 

 may make sure of the same degree of success in an American tele- 

 graph by increasing all the dimensions of the wire in the ratio of 

 the o-reatest distance to which it is to extend, to that for which the 

 French one has been tried." It will be an economical problem, 

 easily solved by the ordinary analytical method of maxima and mi- 

 nima, to determine the dimensions of wire and covering which, with 

 stated prices of copper, gutta percha, and iron, will give a stated 

 rapidity of action with the smallest initial expense. 



" The solution derived from the type Ij- maybe applied to give 



the condition of the wire, when one end, E, is kept connected with 

 the ground, and the other, O, is operated on so that its potential 

 may be kept varying according to a given arbitrary function of the 

 time : only this, which I omitted to mention in my last letter, must 

 be attended to: instead of merely considering sources (so to speak) at 

 O and O' (the latter in an imaginary continuation of the wire), we 

 must suppose sources at O.Oj.Oa.&c, and at 0',0/,02', &c. arranged 

 according to the general principle of successive images, so that the 

 potential at E may be zero, and that at O may be uninfluenced by all 



other sources except the source at O itself. Taking Oj, Oj, O, 



O', O ', O '. . . . equidistant, we have only to suppose equal sources, 

 each represented by the type 



ze *t 



