120  Lord  Rayleigh  on  Electrical  Vibrations 
advance  along  the  series  is  characteristic  of  the  series  of 
spectrum-lines  found  for  hydrogen  and  the  alkali  metals, 
hut  in  other  respects  the  analogy  fails.  It  is  p2,  rather  than  p, 
which  is  simply  expressed  ;  aud  if  we  ignore  this  consi- 
deration and  take  the  square  root,  supposing  n  large,  we  find 
p  cc  1  —  l/2?i, 
whereas  according  to  observation  n2  should  replace  n.  Further, 
it  is  to  he  remarked  that  we  have  found  only  one  series  of 
frequencies.  The  different  kinds  of  harmonics  which  are  all 
of  one  order  n  do  not  give  rise  to  different  frequencies.  Pro- 
bably the  simplicity  of  this  result  would  be  departed  from  if 
the  number  of  electrons  was  treated  merely  as  great  but  not 
infinite. 
The  principles  which  have  led  us  to  (10)  seem  to  have 
affinity  rather  with  the  older  views  as  to  the  behaviour  of 
electricity  upon  a  conductor  than  with  those  which  we 
associate  with  the  name  of  Maxwell.  It  is  true  that  the 
vibrations  above  considered  would  be  subject  to  dissipation  in 
consequence  of  radiation,  and  that  this  dissipation  would  be 
very  rapid,  at  any  rate  in  the  case  of  n  equal  to  unity  *.  But 
this  hardly  explains  the  difference  between  the  two  views  . 
The  problem  of  the  vibration  of  electricity  upon  a  con- 
ducting sphere  has  been  considered  by  Prof.  Thomson  f, 
but  his  solution  does  not  appear  to  me  to  have  the  significance 
usually  attributed  to  it.  For  the  vibration  of  order  1,  the 
value  of  p  (with  the  same  meaning  as  above)  is 
-JU+t} ""• 
But  the  solution  corresponding  thereto  is 
eiptei\r    .  1\ 
Xr 
where  \=p/V,  V  is  the  velocity  of  light,  and  a  the  radius 
of  the  sphere.  Considering  only  the  exponential  factors, 
we  have 
eipteikr  =  e-i(l+iS3XVt-r)/a       ....        (12) 
including  the  non-periodic  factor  e^a.  Thus,  although  (12) 
diminishes    exponentially    with    the    time    and   represents    a 
*  In  this  case  we  should  have  to  consider  how  the  positive  sphere  is  to 
"be  held  at  rest. 
t  Proc.  Lond.  Math.  Soc.  xy.  p.  197, 1884  ;  '  Eecent  Researches/  §  312, 
1893. 
