400 M. G. KirchhofF on the Mution of Electricity in Wires. 



equations distinguished by numbers, under the supposition that 

 the form of the central line of the wire is such, that the distance 

 between two of its points, between which a finite portion of the 

 wire lies, is never infinitely small. By this supposition the case 

 is excluded, that induction spirals are contained in the circuit. 

 In this way we greatly simplify the equations (1) and (3). 



Let A denote the position of the element ds, and B and C two 

 points upon tlie wire, at both sides of A and at a finite distance 

 from it ; then the integral 



extended over the whole wire, with the exception of the piece 



BAC, is a finite quantity, hence infinitely small in comparison 



2e 

 with 2elog — ; hence in the equation (1) this integral must only 



be extended over the portion BAC, with the exception of the 

 portion 2e. Denoting, therefore, by a the arc between A and a 

 variable point of tlie wire, the integral mentioned may be set 



_C^^e^ pcy^^o- 



e* . 

 The quantity — is a function of a, which approximates to the 



g ' 



value - when a approaches ; the integrals 



r('-:)--r(7'-7)- 



have therefore finite values, for the function to be integrated will 

 never be infinitely large ; hence instead of the integral in equa- 

 tion (1) we may set 



that is, 



, AB , AC 



e log h e log — . 



6 ° € 



The choice of the lengths AB and AC is arbitrary, only they 

 must be finite in comparison to the length of the wire ; for both 

 we may set the half of this length : denoting the whole length 

 by /, the equation (1) will be 



that isj 



2e I 



V=2elog — + 2elog-, 



V=2elog-. 



