M. G. Kirchhoff on the Motion of Electricity in Wires. 399 



be neglected in comparison with the member 2iflog — "l )> and 



2e 

 for this we may set 2i log — , if the thickness of the wire be only 



small enough in comparison with the dimensions of the figure 



26 



formed by its axis ; for then e can be so chosen that log — shall 



be a number infinitely great, and e notwithstanding infinitely 

 small in comparison with the dimensions of the figure alluded 

 to. In accordance with this supposition we have 



w=2i\os: 1- 1 i' — cos ^ cos ^', ... (3) 



« J '• 

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

 the exception of the length 2e. 



To the equations (1), (2) and (3), between the four quantities 

 i, e, V, w, a fourth may be added. 



Let two transverse sections be supposed to pass through the 

 commencing and terminal points of ds ; through the first point 

 passes in the time dt into the element of the wire bounded by 

 both, the quantity idt of positive electricity ; through the second 

 point passes in the same time out of the element of the wire the 



quantity of positive electricity yi-\-—ds)dt; the element losesj, 



^i 

 therefore, in the time dt the quantity -^ dsdt of positive electri- 



city ; the negative electricity flows in equal quantity and in the 



opposite direction through both cross sections; the element 



of the wire gains, therefore, in the time dt as much of negative 



electricity as it loses of positive ; its free electricity, that is, the 



difiference between its negative and positive, diminishes therefore 



^i 

 in the element of time \>y 2-^ dsdt: this free electricity is, how- 



ever, eds^ and hence we have 



2^=-!^* (4) 



d« ^t ^ ' 



I will now develope further the theory contained in the four 



* The deduction of this equation is based on the supposition, that in the 

 ease of a non-stationary cuiTent equal quantities of the opposite electricities 

 pass through everj' cross section of the conductor in ef[ual times. If this 

 supposition be, however, rejected, the equations would nevertheless hold 

 good ; it would then be merely necessary to define the intensity of the cur- 

 rent as the arithmetic mean of the two quantities of electricity, which in 

 the unit of time move in opposite directions tlirough the cross section of 

 the conductor. 



