M. G. Kirchhoff on the Motion of Electricity in Wires. 405 



The constants a, A, A' may be determined by the proposition of 

 Fourier, when for ^=0, e is given as a function of s. The solu- 

 tion may, however, be reduced to another form, which shows its 

 characteristics more plainly. 

 For ^=0, let 



e=f{s). 



Let the expressions under the sign of summation in e be modified 



according to the equations 



cosa;siny= ^sm{y + x)+^s'm{y—x), 

 cos a? cos y= i cos (y + a?) + i cos (y — x), 

 Bmxsmy= — icos {y + x) +|-cos {y—x). 



When it is considered that the function / is necessarily periodic 

 with regard to /, we see that the expressions for e and i may be 

 written as follows : — 



.=a + Je-«.[/(.+ ^^>)+f{s- ^0-^4 



••=-4-7l-'[X-73')-/(»-;^0]- 



The quantity a is here determined by the equation 



!=y(/(«)^*J 



that is, la is the quantity of free electricity which the whole wire 

 contains. 



The expression for e shows a very remarkable analogy between 

 the propagation of electricity in the wire, and the propagation of 

 a wave in a tended wire or an elastic rod vibrating longitudinally. 

 When a = 0, that is, when the total quantity of electricity =0, 

 the electricity resolves itself, if I may use the expression, into 

 two waves of equal strength, which run in opposite directions 



through the wire with the velocity -—=. Here the density of 



the electricity diminishes everywhere proportionally with e~'''. 



This diminution, however, in comparison with the velocity of the 



waves, is very slow. The time required by both waves for a revo- 



/ a/2 

 lution is , and hence the ratio of the electric densities at a 



c 

 point before and after the revolution is that of 



c 

 This ratio differs from 1 by an infinitely small quantity, as the 



