OHM OX THE GALVANIC CIRCUIT. 475 



then remains for the determination of the function v, M'hich still 

 possesses the same form as the equation (#), but differs from it 

 in this respect, that v is a function of x, and t of a different 

 nature from ti, by which its final determination is much faciU- 

 tated. 



The mtegral of the equation ( ]) ), in the form in which it was 

 first obtained by Laplace, is 



v = ^-^^^ fe-y' f{x+2y VT't) dy, (?) 



where e represents the base of the natural logarithms, tt the 

 ratio of the circumference of a chicle to its diameter, and / an 

 arbitrary function to be determined from the pecuhar nature of 

 each problem, while the limits of the integration must be taken 

 from j/= — CO toy = + co. For ^ = Ave have v =fx, be- 

 cause between the indicated Hmits/'e~2/" dy ^= \/ -k, whence it 

 results that if we know how to find the function v in the case 

 where y=0, we should thereby likewise discover _/ .z", conse- 

 quently the arbitrary function y. Now in general v = m —«' ; 

 but if we reckon the time t from the moment when, by the 

 contact at the two extremities of the cii'cuit, the tension origin- 

 ates, then M, when ^ = has evidently fixed values only at these 

 extremities, at all other places of the circuit u is = ; accord- 

 ingly, in the whole extent of the circuit v = — ?«' in general when 

 t — Q; only at the extremities of the circuit at the same time 

 ^; = « — v!. If, therefore, we imagine a circuit left from the 

 first moment of contact entirely to itself, then v constantly = 

 at its extremities, so that therefore in the interior of the circuit 

 »=—«', when # = 0, and at its extremities v = 0. Since, in 

 accordance with our previous inquiries, v! may be regarded 

 as known for each place of the circuit, this likewise applies to 

 V when / = ; we know then the form of the arbitrary function 

 fx^ so long as x belongs to a point in the circuit. 

 "• However, the integral given for the determination of v re- 

 quires the knowledge of the function fx for all positive and ne- 

 gative values of x ; we are thus compelled to give, by transfor- 

 mation, such as the researches respecting the diffusion of heat 

 have made us acquainted with, such a form to the above equa- 

 tion that only pre-supposes the knowledge of the function fx 

 for the extent of the circuit. The transformation applicable to 



