385 



" This, if T be infinitely small, becomes 



9v r * 



v= T\ dne~ zn cos(2*-z*) ...... (7), 



"" Jo 



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

 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 : 



r, kg? 



dne~ xn cos(2 zn ) = ^~Tt , 

 4*s 



and when t is negative, =0. 



And so we have 



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



,-Hf *,- 



24) ,_ T S' 



(8), 



or which is the same, 



> I \M\f ,/f n\ / , _\ 



u=r nl - 3 ( .... (10). 



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

 must be taken to t (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 dO 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 : 

 jfl_ 



4 ' Q 



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- 



