408 M. G. KirchhoflP on the Motion of Electricity in Wires, 

 Hence we have to set 



^ = ^^^^""45;^^' 

 and we thus obtain 



e= -— ■< -^^ — e "'Z'ni i- cos WIT sin OTffl ?-. 



4fy II -rr ^ m J 



If the corresponding expression be formed for i, remembering 

 the equation by which it has been defined, we obtain 



j^ cK " C IV". 



i = ( 1 — e"*^*') — -; — ;= — e~*' ^m \ i~ sin mr cos m<6. 



r ^ ' 4A/277r , m 



The meaning of these expressions shall now be developed ; in the 

 first place that of the expression for i. It is our chief object here 

 to find the value of the summation which appears in the expres- 

 sion. We are to regard as a constant, and as a function of t j 

 this function is periodic as regards 27r; it has further opposite 

 values for T and 37r— t; it is sufficient, therefore, to find the 

 values through which it passes when t lies between and tt. 

 We have 



2™ ^ ' sin mr cos »i0 = i 2'" ^^ '— sin »i(t + <^) 



I m " 1 m 



+ 1 '^mS Lsin?n(T — 0). 



, m 



But when x lies between — tt and +7r we have the sum 



8m»na?= — — , 

 \ m 2 



and because it is periodic as regards 27r, it is in general 



= -i(a?-2/w), 

 where JO denotes that whole number, for which x—^pir lies be- 

 tween — TT and +7r. With the limits which have been assumed 

 for T, T— ^ lies always between — tt and +7r, because for all 

 points of the wire the value of ^ is between and tt. Hence 

 we have 



i — VT • , .\ "r— 

 2 ^^— ^ • sin »j(t- (/>) = -7j^. 



With regard to the value of t + <^, this can be either greater or 

 less than tt. We have 



2^ '^va.m{T-^'^)=- — '^—^, when<^<7r— T, 



m 



2 

 — ^ +7r, when 0>'7r — t. 



