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

 thus obtain 



g—Q-hiQQ^ — =(Asin«s + A'cos?w), 



c ,, . cnt 



r=e 



-M 



sin — = (A cos ns — A' sin ns). 



A particular solution of another form, which also satisfies the 

 condition that for t = 0, i vanishes, is 



where a and b denote two arbitrary constants. That the two 

 differential equations are satisfied by this is easily seen, by ob- 

 serving that by introducing the quantity h, one of them assumes 

 the form 



V4 ds •&// 



A solution is obtained which can be made to agree with the 

 further conditions of the problem,'where e and i are made equal 

 to the sums of partial solutions of the forms stated. 



"We shall now examine more particularly the case in which the 

 wire is one returning into itself. In this case e and i must have 

 equal values for s = and for s = l; and this must moreover take 

 place whatever the origin of s may be ; this requires that e and i 

 are functions of s, which are periodic with regard to I ; for this 

 it is necessary that 



27r 

 i=0 and n=m—j-, 



where m denotes an integer. We have thus for e and i the fol- 

 lowing expressions : — 



i,v« A 27r c . . 27r 

 g_g-Ai^mA,^cosm-j -=^.smm-^ s, 



it^ ki Stt c ^ 27r 



+ a + e~''' i-"* Am cosffi -, -= t .cosm-j-s, 



i = — - — -=e~*'2.»nAmSin7w^ -=t .co?>m—rS, 



2^2 , I V2 I 



c i/-^ Ai • 27r c , . 27r 

 + 9 • /- e~''^Z'^A'm^mm-: — j=t,smm-j-s. 



