Fields  by  Means  of  an  Oscillating  Electric  Needle.       403 
ellipsoid  the  couple  is  proportional  to  the  sine  of  twice  the 
angle  6  between  the  long  axis  and  the  field.  Assuming  that 
law  to  hold  for  a  cylindrical  needle,  we  may  write  the  couple 
equal  to  a~F2  sin  20,  where  a  is  some  constant.  If  6  is  small 
we  have 
, ,   C°up!e  "  =  aF'™2g=2qF»  (constant) , 
Angular  displacement  u 
in  which  case  the  vibrations  of  the  needle  will  be  isochronous, 
the  time  being  given  by 
T=2V5 ^ 
where  I  =  moment  of  inertia  of  the  needle. 
Writing  T  =  ^ ,  we  may  denote  the  frequency  of  vibration 
by  N=6F, (II) 
where  b  is  a  constant  depending  on  the  form,  size,  and  mass 
of  the  needle.  Thus  it  appears  that  the  strengths  of  uniform 
fields  are  directly  in  proportion  to  the  frequencies  of  such  a 
needle  oscillating  in  them. 
Of  course  if  the  controlling  couple  due  to  the  suspending 
fibre  is  sufficient  to  take  into  account,  this  may  be  done  by 
observing  the  frequency  N0  when  the  electric  field  is  nil. 
Denoting  the  frequency  actually  observed  in  field  by  N', 
we  have 
N=  VN'^-N.'-'^^F (Iiy 
This  law  was  tested  by  establishing  an  electric  field 
between  a  pair  of  parallel  circular  plates,  each  12  cms.  in 
diameter,  kept  at  a  constant  distance  apart.  The  difference 
of  potential  between  them  was  obtained  by  means  of  a  Kelvin 
voltaic  pile,  whereby  any  voltage  from  400  volts  downwards 
could  be  applied.  The  volts  were  measured  by  a  Kelvin 
multicellular  electrostatic  voltmeter  reading  to  400  volts. 
The  vibrations  were  observed  by  the  help  of  a  plane  mirror 
placed  below  the  needle,  as  shown  in  fig.  1,  which  represents 
the  apparatus  used.  The  image  of  the  needle  in  the  mirror 
was  viewed  through  a  telescope,  in  the  focal  plane  of  which 
an  image  of  the  needle  appears  vertical  in  the  position  of 
rest.     The  vertical  cross-wire  is  set  to  bisect  it. 
The  time  of  a  counted  number  of  oscillations  was  obtained 
by  means  of  a  stop-watch  reading  to  fifths  of  seconds, 
sufficient  passages  being  allowed  to  admit  of  the  whole  period 
of  time  measured  being  from  one  to  two  minutes. 
