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



and hence ,r „ , 2e Ce'd^ ,, v 



V=2elog— +J^^, . . . . (1) 



where the integration is to be extended to the whole wire, with 

 the exception of the portion 2e. 



We have now to form the expression for the electromotive 

 force induced in the point under consideration, by the alteration 

 of the intensity of the current in all portions of the wire. 



"When in the element of a conductor, the length of which is 

 I', the intensity of the current denoted by i' changes, an electro- 

 motive force will be induced by this change in a second element 

 of the conductor, which, with reference to the unit of electricity 

 of Weber*, is 



8 di' I' . ., 

 = 5- 7— - cos ^ . cos 6', 



c^ 3/ r 



where 6 and ^' denote the angles formed by the two elements 

 with the line drawn from the first to the sccoiid, r tlie length of 

 this line, and c the constant velocity with which two particles of 

 electricity must move towards each other, so that they may exer- 

 cise no force upon each other. 



For all parts of the wire, excepting the piece of the length 26 

 already alluded to, the electric current may be regarded as con- 

 centrated in tlie central line : the portion of the induced electro- 

 motive force now sought, which is derived from the wire, with 

 the exception of the piece already mentioned, is therefore 



where i' is the intensity of the current which passes through the 

 cross section of the wire at the place r/i', Q and ^' the angles 

 which the elements ds and db' form with the liiae which is drawn 

 from the latter to the former, r the length of this line, and where 

 the integration is to be extended throughout the whole wire, with 

 tlie exception of the portion already referred to. 



In this portion the current must not be regarded as concen- 

 trated in the central line, but in lieu of this it may be regarded 

 as straight and parallel to ds. Through the first point of rfs' let 

 a transverse plane be placed cutting the wire, and let p' and A|r' 

 be the polar coordinates of a point of the plane, with reference 

 to a system of coordinates whose origin is the centre, and whose 

 axis is parallel to the line from which the angle -^ is reckoned : 

 if then the density of the current in the points determined by p' 

 and ■>^^ be J', we obtain for the portion of the induced electro- 



* Elektrodynamische Massbestimmungen, 1846, p. 354 ; and 1856, p. 268. 



