410 M. G. Karchhoff on the Motion of Electricity in Wires. 

 nitude of the break is 



or if we denote by J the vahie to which i approximates as the 

 time is increased, that is, the value of — , 



= J 



8 a/27 



This quantity has its greatest value when /=0; but this, in ac- 

 cordance with an assumption already made, is also infinitely small 

 in comparison to J. The expression for the magnitude of the 

 break may be more shortly written, when the time is introduced 

 required by the point at which it takes place, or the time required 

 by an electric wave to move through the length of the wire. 

 Denoting this time by T, that is, setting 



^~ c ' 



the expression is easily found to be 



= J.2ATe-''^ 



As the time increases, the magnitude of the break diminishes, 

 but so slowly that during the time T only an infinitely small 

 diminution takes place. 



To obtain a complete view of the process, it is now only ne- 

 cessary to examine the alterations of the strength of the current 

 at the commencement of the wire. Let this, that is to say, the 

 value of i for *=0, be Jq; then making use of the symbols J 

 and T, we find 



«„ = J(l -e-2''') + ^i^e-"'i«.iz:Il!' sin mr. 

 71" , m 



Setting for the sum its value, and remembering that 



7r~T' 

 we obtain 



io=H'i -e-"")+32he-'"{2pT-t), 



where p denotes the whole number for which 

 t-2pT 

 T 

 is a proper fraction, positive or negative, p may also be defined 

 as the greatest integer which is contained in the fraction 

 t_+T 

 2T ■ 



