Prof. Thomson on the Electric Telegraph. 149 



an infinitely short time in communication with the battery and 

 immediately after with the ground. It may be reduced at once to 

 finite terms by the evaluation of the integral, which stands as fol- 

 lows : — 



when Hs positive, I dne~''"' cos(2nt—zn) = — ^e u , 



J n 4ft 



and when t is negative, =0. 



And so we have 



4.^ti 

 or by (6), when t is not infinitely small. 



«'=T-^e ^' (8), 



or which is the same. 



VzC dd 

 2kV t-Td^ 



2rrVo(t- 



de 



\(t-d)^ 



(9). 



(10). 



It is to be remarked that in (9) and (10) the limits of the integral 

 must be taken to f (instead of t — T to t, or to T), if it be de- 

 sired to express the potential at any time t between and T, since 

 the quantity multiplied by dd in the second number of (6) vanishes 

 for all negative values of 6. 



" These last forms may be obtained synthetically from the follow- 

 ing solution, also one of Fourier's elementary solutions : — 



-F-7V^ <">• 



which expresses the potential in the wire consequent upon instan- 

 taneously communicating a quantity Q of electricity to it at O, and 

 leaving this end insulated. For if we suppose the wire to be continued 

 to an infinite distance on each side of O, and its infinitely distant ends 

 to be in communication with the earth, the same equation will ex- 

 press the consequence of instantly communicating 2Q to the wire 

 at O, Now suppose at the same instant a quantity — 2Q to be com- 



muuicated at the point O' at a distance — r^ on the negative side 



\^kc 



of O, the consequent potential at any time t, at a distance ^_ 



along the wire from O, will be , 



"=?{-? — r-\ <'^'' 



