Erof.  M.  B.  Pell  on  the  Constitution  of  Matter.         171 
depend.  If  any  system,  not  subject  to  loss  of  energy  by  friction 
or  any  similar  cause,  be  vibrating  about  a  position  of  stable  equi- 
librium, the  average  proportion  of  the  energy  in  a  potential  state 
depends  greatly  upon  whether  or  not  there  is  any  position  of 
unstable  equilibrium  within,  or  nearly  within,  the  scope  of  the 
motion.  When  the  system  passes  slowly  through  such  a  posi- 
tion, a  large  proportion  of  the  energy  becomes  potential ;  and  if 
the  motion  constitutes  heat,  a  large  proportion  of  the  heat  be- 
comes latent.  The  existence  of  the  position  of  unstable  equili- 
brium, B,  appears  to  be  the  chief  cause  of  the  various  changes 
which  are  observed  in  the  specific  heats  of  solids,  gases,  and 
vapours. 
Let  there  be  n  equal  atoms  in  a  straight  line  acting  upon  one 
another  according  to  the  law  already  assumed,  and  slightly  dis- 
turbed in  that  line  from  their  positions  of  stable  equilibrium. 
Let  the  displacements  at  time  t  be  represented  by  xif  x2) . .  •  xr. . . 
The  equations  of  motion  to  a  first  approximation  are 
1   d*x,  __ 
1  d*x 
dP 
—j-£  -\-2x2—xi  +  x3, 
or,  putting  ?  for— 2(^-)  +2, 
qxr  =  xr-i-\~xrjrl) 
> a) 
[q— l)#»=#„__i. 
Let  these  equations  be  multiplied  by  fv  /2, . .  .fr  . . .  respectively 
and  added  together  j  and  let  all  the  terms  disappear  from  the 
resulting  equation,  except  that  involving  xn.  We  must  have  then 
This  gives  us 
/r=Acos(r0  +  B), 
where 
q=2cos0. 
The  condition  that  xl  disappears  is 
(2  cos  0 -!)/,=/„ 
