-M. 6. Kirchhoff on the Motion of Electricity in Wires. 407 



wire at the time ^=0. We may then assume that at the com- 

 mencement of the wire, or when 5 = 0, the potential is always 

 = 0; and at the end of the wire, or for s = /, it has a constant 

 value depending upon the electromotive force of the circuit- 

 When K denotes the electromotive force, this value must be iK. 

 The conditions to be satisfied by the expressions foi* e and i 

 are therefore the following : — 



for s=0 we must have e=0, 



... s = / ... e= — - K, 



4y 



... ^=0 ... e=0. 



On account of the first condition, we must have the quantities 

 A'=0 and also a=0. As for s = /, e is to be independent of t, 

 the quantities n must satisfy the condition 



sinn/=0; 

 that is, we must have 



TT - 



n=in-Y, 



where m denotes a whole number. Further, in order that for 

 s = l, e shall have the required value, we must make 



'iyl 



Setting, for the sake of shortness, 



TT c ^ 





and 



/ 

 we obtain for the equation 



K ■" 



e= - — , s + e~''' 2»» A«,co8 mr sin md). 



The constants A may be determined by the last condition : 



according to this, for all values of ^ between and tt we must 



have 



K °° 



:; — <f> = — 2'" A,,, sin md>. 

 477r , ^ 



But, by Fourier's proposition, between the same limits we have 

 the following equation : — 



</) = - 2 2"' ( - 1 r — sin /«</>. 



