148 Royal Society : — 



"Eliminating y from (1) by means of this, we have 



cA^=^, (3), 



dt dor ^ ' 



which is the equation of electrical excitation in a submarine telegraph- 

 wire, perfectly insulated by its gutta percha covering. 



" This equation agrees with the well-known equation of the linear 

 motion of heat in a solid conductor ; and various forms of solution 

 which Fourier has given are perfectly adapted for answering practi- 

 cal questions regarding the use of the telegraph-wire. Thus first, 

 suppose the wire infinitely long and communicating with the earth 

 at its infinitely distant end : let the end O be suddenly raised to 

 the potential V (by being put in communication with the positive 

 pole of a galvanic battery of which the negative pole is in commu- 

 nication with the ground, the resistance of the battery being small, 

 say not more than a few yards of the wire) ; let it be kept at that 

 potential for a time T ; and lastly, let it be put in communication 

 with the ground (t. e. suddenly reduced to, and ever afterwards 

 kept at, the zero of potential). An elementary expression for the 

 solution of the equation in this case is 



^_Vr°° -^»^ sin \_int-zrt^ - sin [(^-T)2«-ga^] 



yJT 



TT I 

 c/0 



■where for brevity 



2=x \/ kc (5)." 



That this expresses truly the solution with the stated conditions 

 is proved by obsen'ing, — 1st, that the second member of the equa- 

 tion, (4), is convergent for all positive values of z and vanishes 

 when z is infinitely great; 2ndly, that it fulfils the differential 

 equation (3) ; and 3rdly, that when 2=0 it vanishes except for 

 values of t between and T, and for these it is equal to V. It is 

 curious to remark, that we may conclude, by considering the phy- 

 sical circumstances of the problem, that the value of the definite 

 integral in the second member of (4) is zero for all negative values 

 of t, and positive values of z. 



" This solution may be put under the following form. 



«=— I dd\ 



""Jt-T Jo 



(/«€"*"* cos(2«0-^n*) . . . (6)." 



which is in fact the primary solution as derived from the elementary 



/ it 7ri\ /^ 



type cos ( 27!"??; — ^ ^^t) ^~'' "^ given by Fourier in his investiga- 

 tion of periodic variations of terrestrial temperature. 

 " This, if T be infinitely small, becomes 



v=.— i:\ rf«e-^"* cos(2n^-r«*) (7), 



'^ Jo 

 which expresses the eflfect of putting the end O of the wire for 



