182         Prof.  M.  B.  Pell  on  the  Constitution  of  Matter. 
Let  fjb  =  2m  sin  yjr;  then  6}  operating  upon  cos/z/,  is  equal  to 
2a|t,  and 
Xa  sin  2{n—rs)^r  cos  fit 
r  sin  2)iTJr 
Now  X,  representing*  the  ratio  of  the  action  of  an  atom  of 
sether  to  the  mutual  action  of  the  atoms  of  a  solid  body,  is  very- 
small — almost  infinitesimal.  The  above  coefficient  of  cos  fit  is 
therefore  wholly  insignificant  except  when  sin  2n-^r  is  very  small, 
or  when  2n-ty  =  a  multiple  of  nr.  In  this  case,  as  before  ex- 
plained,  the  quantity  by  which  X  is  multiplied  is  large,  but  not 
infinite ;  for  as  the  amplitude  of  the  vibration  increases,  the  time 
of  vibration  is  slightly  increased,  as  indicated  by  the  factor  c  intro- 
duced in  the  second  approximation ;  and  ijr  is  thus  slightly  dimi- 
nished. The  only  values  of  ty  which  produce  any  sensible  effect  are 
it    27T  Sir   p  T    ,    ,       vir  , 
2n'  2n    %i  ^=  2n  =  U%  'C  V  1S  an  mteSer  '> 
putting  b  =  -—. -^  we  have,  supposing  the  n  atoms  to  have  been 
r         °        2  sm  nd  rr        ° 
initially  at  rest  in  their  positions  of  stable  equilibrium, 
lV  5cos(2»-2r  +  l)7 
xr — a1  +  Z  ras  cos  at ^ —  cos  at. 
r  s  sin  vy  n ' 
where  al  ras  are  arbitrary  constants,  and  fis  =  2m  sin  sy.     Putting 
„  ~  6cos  (2n— 2r  +  l)vy 
1         r  sin  vy 
Equation  (8)  gives 
cos  (2n— 2r  +  lW 
rGs     ns  cos  sy 
whence  it  may  be  shown  that  nas,  and  consequently  ras)  vanishes 
for  all  values  of  s  except  s  =  v,  and 
nCly  b 
cos  vy      sin  vy 
&cos(S»-2r+l)vy  o 
sm  vy  v  J 
Z>cos(2/z-2r+l)^  . 
= : : —    (1  —  COS  tit)  . 
sim|r  V  ^  ' 
If  the  disturbance  be  supposed  to  cease,  it  may  be  shown,  as 
before,  that  the  subsequent  motion  is  represented  by 
2>cos(2n— 2r+lM 
ccr= = — : s_L_  cos  at : 
sim/r  n 
so  that  the  system  radiates  the  same  kind  of  heat  which  it  ab- 
sorbs.    If  /jl  be  greater  than  2m,  xr  is  small,  even  in  comparison 
