Prof.  M.  B.  Pell  on  the  Constitution  of  Matter.         175 
Since  2mr=ry,  it  is  evident  that 
%n— r+l=       %r  > 
so  that  the  motion  is  symmetrical  about  the  middle  point.  If  n 
be  an  odd  number  the  central  atom  will  remain  at  rest.  These 
conclusions  hold  for  the  higher  approximations  as  well  as  for  the 
first. 
The  whole  energy  is  -^—  ;   and  the  vibratory  energy,  which 
71  —  1 
alone  is  represented  in  the  above  value  of  xr,  is  ~ —  mPa*.     The 
in 
heat  developed  as  temperature  for  the  rth  atom  is 
mVv   .  9       .  Q/0       ,N         ??zV(/i  — 1) 
-^-2sin««7  8ma(2r-l)«7=  —^ L, 
and  is  the  same  for  all,  the  mass  of  an  atom  being  here  supposed 
to  be  unity. 
Subject  to  no  external  disturbance  whatever,  any  number  of 
atoms  might  vibrate  together  in  the  manner  indicated;  but  this 
is  a  condition  which  cau  never  exist ;  for  the  atoms  of  a  molecule 
of  vapour,  even  during  the  interval  between  their  encounters,  are 
subject  to  acceleration  or  retardation,  as  the  case  may  be,  from 
the  action  of  the  aether  in  which  they  must  be  supposed  to  be 
immersed.  A  notion  may  be  formed  of  the  effect  of  an  external 
disturbance  upon  such  a  system  as  that  under  consideration,  by 
supposing  an  additional  atom  at  the  beginning  of  the  series  con- 
strained to  move  according  to  a  particular  law.  Let  xQ  be  the 
displacement  of  this  atom,  and  suppose 
xQ  =  acosfxt} 
a  being  small  compared  with  h.     The  equations  of  motion  are 
qx^=xQ  +  #2, 
qxr=xr+l-\-xr+1) 
(q~l)xn  =  xn.1} 
1    /d^ 
where,  as  before, 
*=2+iW)=2cos(?- 
If  these  equations  be  multiplied  by  fv  /2, . . .  respectively  and 
added  together,  and  all  the  terms  disappear  from  the  resulting 
equation  except  those  involving  x0  and  xn,  we  have 
2cos^/'r=/r_1+/r+1, 
which  gives/  =Asin  (r#-f  B). 
