176         Prof.  M.  B.  Pell  on  the  Constitution  of  Matter. 
The  condition  2/,  cos  9=f2  gives  B  =  0;  so  supposing /j  =  l, 
we  have 
-  ___  sin  r6 
and  the  coefficient  of  scn  is 
(g°°»g-iy.-/.-.=  CoSig  • 
The  equation  is  therefore 
cos(n-j-J)0 
in      ocn—a  cos  /a*. 
cos  J0        n  r 
We  have  also 
a?y=Acos  (r$  +  B)  ; 
the  condition  (2cos#- 
that 
and  the  condition  (2cos#— l)%n=%n-i  gives  B=  — - — Q;  so 
#V=Acos  (n— r-\-\)9, 
cos(rc-r  +  i)<9 
*r~         cos|0        ^ 
_    cos  (n— r  +  J)0 
—  -  ,        iv/,  cos^. 
cos(?z-f-f)0        ^ 
If  we  put  /u,=2?rasimjr,  then,  when  operating  upon  cos /it  f 
0=2^,  and 
cos(2w— 2r  +  l)i|r 
,rr=  a ^—jf. — ,  -,  .,      cos  fit, 
COS  (2w  +  l)^r  ^ 
The  tendency  to  rupture  is  a  function,  not  of  the  displace- 
ments, but  of  the  relative  displacements  of  the  atoms,  repre- 
sented by 
2a  sin  (2n  —  2r  +  2)  ylr  sin  yjr 
xr—xr-\  = 7^ — ,  -,;  . cos  at, 
r  COS  (2ft  +  l)l/r  ^ 
If  n  be  considerable,  but  yjr  so  small  that  (2w  +  l)^  is  not 
TT 
nearly  equal  to  — ,  sin  ty  will  be  small,  and  the  relative  disturb- 
ance small  compared  with  a  for  all  values  of  r.  This  shows  that 
a  slow  external  disturbance/ corresponding  to  a  small  value  of  fi, 
will  cause  a  general  oscillatory  motion  of  the  whole  system,  but 
very  little  internal  relative  vibration  of  the  atoms. 
*  TT 
If  (2*1  +  1)^  be  equal  to  — ,  or  to  any  odd  multiple  of  it, 
#r— -#V-i  becomes  infinite,  indicating  the  well-known  change  in 
the  form  of  the  solution  from  A  cos  fit  to  At  sin  fit.  I  will  defer 
the  consideration  of  this  particular  case,  and  suppose  (2n-\- 1)^ 
