128  Prof.  W.  Weber  on  Electricity  in  relation  to 
0=^/^  +  2"o«o) 
(cc 
^MV(>+£)W(*^))> 
or,  for  small  values  of  -$> 
c 
8-?(i-^lO 
If  we  next  confine  ourselves  to  the  consideration  of  small  oscil- 
r 
lations  (that  is  to  say,  those  for  which  —  is  very  small),  it  results 
from  the  above  equation,  when  r0  and  r  are  taken  as  vanishingly 
small  compared  with  p,  that 
P         i-  \ra        ?   t 
dv 
whence,  putting  u=  -r-> 
at 
cdt 
=  —  dr  x  / , 
V  Vo¥o  |  r   Y1    |  Wff 
pec  \r0       pec  J 
which  leads  to  an  elliptic  integral.     For  vanishing  values  of  — , 
we  obtain 
cM=-dr*J -; 
whence  there  comes,  by  integration, 
-if0 
°Jr0 
&:---»   - * ^a. 
v/('-9 
When,  as  has  been  assumed,  r  is  <  p}  r0  may  be  called  the 
amplitude  of  oscillation ;  and  it  follows  that,  for  small  values  of 
« 
—  and  for  small  amplitudes  of  oscillation,  the  time  of  oscillation 
2©  of  an  electrical  atomic  pair  is  proportional  to  the  amplitude 
of  oscillation  r0.    But  the  factor  with  which  r0  must  be  multiplied 
4 
in  order  to  give  20,  though  a  constant  =  -  for  small  amplitudes, 
2 
diminishes  for  greater  amplitudes,  and  becomes  =  -  for  the  am- 
c 
plitude  r=p. 
