M. G. KirchhoiF on the Motion of Electricity in Wires. 411 



For values of t, for which the number p is not very great, the 

 expression for i^ is capable of a considerable simplification. For 

 such the quantity ht is infinitely small ; and by neglecting mem- 

 bers of higher orders, the equation for /q may be thus written : 



iQ=i .2ht + i2h{2pT-t\ 

 that is, 



io=ioJ4AT. 



This expression shows that the intensity at the commencement 

 of the wire is up to the time when / = T ; here and at the times 

 < = 3T, / = 5T, &c., it alters itself by jumps; and moreover the 

 jump is twice as great as at other points of the wire. During 

 the intervening times the intensity is constant. 



In a similar manner the expression for e may be discussed. 

 We have 



2»n .^^ '— cos ruT sin m<^ = ^ "^m \ L. sin m(T + </>) ; 



\ m I m 



or as soon as t lies between and tt, 



-_^ 



~ 2' 



when (f) <r tt— t 



TT 



= — ^ + -g , when <p > TT — T ; 

 if T lies between tt and 2??, we have the same sum, 



= — ^, when0<T — TT 



~~ 2 "*"!' ^^^" <^>T-7r. 



The second fact follows from the first, when it is considered that 

 the sum has the same value for t and for 27r— t. For greater 

 values of t, the value of the sum is found when we remember 

 that it is periodic with reference to 27r. 



From this it follows that at each moment at some one point of 

 the wire, e also suffers a break. This point always coincides with 

 that in which the break for i takes place, e is always gi-eater 

 on the side of this point on which the end of the wire lies, and 

 smaller on the side of the commencement. The magnitude of 

 the break is 



~4y ' 



or, denoting by E the constant value of e at the end of the wire, 



= Ee-«. 

 At that side of the break on which the commencement of the 



2F2 



