M. G. Kirchhoff on the Motion of Electricity in Wires. 409 

 From this it follows that 



( — 1)"' T 



2-^ ^ cos mr cos?«0 = — -i^, when 6 <7r— t, 



7n <* 



T TT 



= — ^ + -^, when (f) > 77 — T. 



It is here supposed that t lies between and tt ; if it lies between 

 •TT and Stt, we have the same sum. 



and 



= 7r — ^, when < t — tt, 



TT T 

 =jr — -^, when ^ > T— TT. 



To find the sum for greater values of t, it is to be remembered 



that it is periodic as I'egards 27r. 



From this it appears, that at every moment a point exists in 



the wire in which the intensity of the current suffers a sudden 



change or break. This point, at the time / = 0, lies at the end 



c 

 of the wire, but moves from this with the velocity —= towards 



the commencement, after reaching which it returns with the 

 same velocity towards the end; turns here again, and thus tra- 

 vels perpetually to and fro over the length of the wire. In each 

 of the two portions into which the wire is at each moment divided 

 by this point, the same intensity exists everywhere at that mo- 

 ment ; so that if s and i be regarded as rectangular coordinates 

 of a point, a line is described of the form of fig. 1 . The inten- 

 sity before the point at Fig. 1. 

 which the break occurs, 

 considered without re- 



gard to its sign, IS always i 



the smaller, that behind 



thepointthegreater,the ^ ■ — 



words before and behind 

 being used with refer- 

 ence to the direction in which the point moves. The figure 1 

 is therefore true only Fig. 2. 



for a moment in which 1 



the point moves from ' 



the end towards the 



commencement of the . , 



wire. The figure 2 re- 

 fers to a moment in which the opposite takes place. The mag. 



Phil. Mag. S. 4. Vol. 13. No. 88. June 1857. 2 F 



