and  the  Constitution  of 'the  Atom.  119 
disturbance.     It  is  worthy  of  note  that  the  constant  (yp0)  of 
itself  determines  a  time. 
In  considering  the  significance  of  the  vibrations  expressed 
by  (6),  we  must  remember  that  when  s  is  uniform  no  external 
forces  having  a  potential  are  capable  of  disturbing  the 
uniformity. 
We  now  pass  on  to  vibrations  not  involving  a  variable  s, 
that  is  of  such  a  kind  that  the  fluid  behaves  as  if  in- 
compressible. An  ir rotational  displacement  now  requires 
that  some  of  the  negative  fluid  should  traverse  the  surface  of 
the  positive  sphere  (a).     In  the  interior  V2R  =  0. 
To  represent  simple  vibrations  we  suppose  that  c£,  &c.  are 
proportional  to  eipt.  By  (3)  V2<jb  =  0;  and  we  take  (at  any 
rate  for  trial) 
<£  =  ^V*SB (7), 
where  S»  is  a  spherical   surface  harmonic  of  the  nth  order. 
The  velocity  across  the  surface  of  the  sphere  at  r  —  a  is 
d<j>ldr=nan-ltfpiS» ; 
and  thus  the  quantity  of  fluid  which  has  passed  the  element 
of  area  da  at  time  t  is 
p[d±dt.dc=P^e*$J*       .     .     .     (8). 
r  J  dr  ip  "  v  J 
The  next  step  is  to  form  the  expression  for  E,  the  potential 
of  all  the  forces.  In  equilibrium  the  positive  and  negative 
densities  everywhere  neutralize  one  another,  and  thus  in  the 
displaced  condition  R  may  be  regarded  as  due  to  the  surface 
distribution  (8).  By  a  well-known  theorem  in  Attractions 
we  have 
4^0nr"S,g'* 
ip(2n  +  l)  K  J 
But  by  (3)  this  is  equal  to  d<f>Jdt,  or  ipe^r"^.  The 
recovery  of  rn$n  proves  that  the  form  assumed  is  correct  ; 
and  we  find  further  that 
.4*7  Po- n (10) 
This  formula  for  the  frequencies  of  vibration  gives  rise  to 
two  remarks.  The  frequency  depends  upon  the  density  p0, 
but  not  upon  the  radius  (a)  of  the  sphere.  Again,  as  n  in- 
creases, the  pitch  rises  indeed,  but  approaches  a  finite  limit 
given  hy  jr  =  '27ryp{y     The   approach   to  a  finite   limit    as   we 
