M. Gr. Kirchhoff on the Motion of Electricity in Wires. 395 



account of its infinitely small length it may be regarded as 

 straight, the quantity of V derived from it is 



27rJ_^J„ \/^-'2 + a2 + p2-2apcos(i/r'— ^)" 



In this expression a^ has been written for s' — s, and y^ denotes 

 the angle between the radius drawn t© an element of the surface 

 of the wire and the line from which the angle i^ is reckoned. 

 "WTien the integration, according to a', is carried out, e, in com- 

 parison with a and p, being regarded as infinitely great, we have 

 the following expression : — 



e r^" 



= -J^ ^•>|r'(log2e-log \/a2^p2_2apcos('»/r'-'»^)), 

 that is, 



= 2e(log26- ^1 <;^'log v/a2 + p^-2a/)COS (^'-A|r)). 

 Setting 



I fi?>/r'log ^/a2_,.p2_2«pcos (-f' -'f)=U, 

 the difici'ential equation 



must be satisfied, because the quantity under the sign of inte- 

 gration multiplied by rfi/r' satisfies this equation for all the values 

 of ilr'; but it is easily seen, by setting yjr'—ylr instead of i/r' as 

 the variable according to which the integration is to be carried 

 out, that U is independent of t|r ; hence we must have 



dp^ p dp~ ' 

 but from this it follows that 



U = C,logp-FC2, 

 where Cj and Cg denote two unknown constants. These may 

 be easily determined by assuming p as infinitely small in com- 

 parison with « ; the carrying out of the integration in the ex- 

 pression for U gives then 



U=27rlog«, 

 from which it follows that Cj is =0, and U has this constant 

 value for all the values of p. Consequently the portion of V 

 derived from the piece 2e of the wire is 



=2eIog-^, 

 2E2 



