328 Mr Pocklington, Electrical Oscillations in Wires, [Oct. 25, 

 Now, n being an integer, 



■K 



2 f2 



d-yjr cos 2nijr cos (x sin yjr) = J., n (x), 



7T Jo 



TV 



% [ 2 j i of sinOsin-v/r) t* ■ 



- avr cos znvr \ — — — = I ax J. m (x) 



vrJo r smf J 



= - 2 [ Ja^-! 0) + Jsn-s (a?) + . . . + J i (x)] + dx J (x). 



Jo 



Therefore the first integral in (5) is 



s {" I J - w + ( 2 - i ) J - w + (! - ! ) />'■<•>} - 



and (4) becomes, on re-arrangement and multiplication by a, 



(X 2 -\ T X 2 _ LIT { X' 2 T . . 



+ (2 - 1) j", w + [j - i)/ o * *» /.(«)} 



(a? 2 P 7 , cos (aj sin ilr) - 1 _ IN 



— Itt ^ *— * — r-^ (1 — 2 cos 2i/r + cos 4iir) 



(8 Jo sinf T T/ 



IT 



(x 2 ,\ [ 2 7, cos (x sin olf) - 1 „,} , ns 



+ U - Vi. # S in7 " cos 2 *j - (6) - 



This equation gives # and thence a with an error of the order 

 of e/a. It can only be solved by trial. If however e is so small 

 that errors of the order of 1/L 2 can be neglected, we may use an 

 approximate solution of the above. A first approximation is x 2 = 4 

 or x = 2. A second is obtained by putting x = 2 on the right- 

 hand side of (6). This gives 



^-l)i = -485--703t, 



so that 



a = i{l+(-243--351t)/i}- 



Qj 



Hence the period of the oscillation is equal to the time re- 

 quired for a free wave to traverse a distance equal to the circum- 

 ference of the circle multiplied by 1 — '243/ L, and the ratio of the 

 amplitudes of consecutive vibrations is 1 : e~ 2 ' 21/L or 1 : 1 — 2 21/L. 

 It is easy to verify from first principles that the decrease in 

 amplitude of the vibrations is of this order. 



