258 The Mathematical Electricians of the 



The next factor to be considered is the mutual induction of 

 the current-elements in different parts of the wire. Assuming 

 with Weber that the electromotive force induced in an element 

 ds due to another element ds' carrying a current i' is derivable 

 from a vector-potential 



,.3 



Kirchhoff found for the vector-potential due to the entire wire 

 the approximate value 



w = 2i log (//a), 



where i denotes the strength of the current ;* the vector- 

 potential being directed parallel to the wire. Ohm's law then 

 gives the equation 



ldw 



where k denotes the specific conductivity of the material of 

 which the wire is composed; and finally the principle of 

 conservation of electricity gives the equation 



di _ _de . 



dx~ ~di' 



Denoting log (I/a) by y, and eliminating e, i, w from these four 

 equations, we have 



8 2 F 1 d*V 1 8F 



which is, as might have been expected, the equation of telegraphy. 

 When the term in 3 V/dt is ignored, as we have seen is in certain 

 cases permissible, the equation becomes 



8 2 F lF 



* This expression was derived in a similar way to that for F, by an intermediate 

 formula 



2 c ci'ds' 



w = 2i log -- h cos 6 cos Q , 

 & a J r 



where 6 and Q' denote respectively the angles made with r by ds and ds'. 



