M. G. Kirchhoff on the Motion of Electricity in Wires. 397 

 motive force due to the portion of wire 2e, the expression 



""^Jo Jo J-. ^ ■ (^^n + p^ + p>^-2pp' cos (t'-t))*' 

 As J' may be regarded as independent of x', the integration 

 according to x may be easily accomphshed : making use of the 

 fact that e is infinitely great in comparison with all values of p 

 and p', we obtain 



""?! ) ^p'dp'dylr'[\og2e-l-\og^/p^ + p'^-2ppcos{ylr'^f)-]. 



r 



i'p' dp' d-\lr = i, 



But as 



this expression is 



Jo Jo 



16 r /I 





-\ \ -^p' dp' df log Vp^+p'^'~2pp' cos{^'-f)j- 



Hence the entire induced electromotive force is- 

 _8 dW 



where 



W= G— co3^cos^' + 2i(log2e-l) 



a /vir 



-2» \ J'p'rfp'<ZA/r'log\/p2 + p'2-2pp'cos(^'-t). 



Jo Jo 



In the case of a stationary electric current, the density of the 

 current is equal to the product of the electromotive force, referred 

 to the unit of quantity of electricity, and the conductivity ; I will 

 assume that the same also holds good when the current is not 

 stationary. This assumption will be fulfilled when the forces 

 acting upon the electricity, and which constitute the resistance, 

 are so powerful that the time during which a particle of electri- 

 city remains in motion after the cessation of the accelerating 

 forces, and in virtue of its inertia, may be regarded as infinitely 

 small, even in comparison with the small space of time which 

 comes into consideration in the case of a non-stationary electric 

 current. According to this assumption, if k be the conductive 

 capacity of the wire, J the density of the current at the point 

 determined by the values of s, p and -f at the time /, we have 



