180         Prof.  M.  B.  Pell  on  the  Constitution  of  Matter. 
Sfjb  represents  the  range  of  fju  in  the  prevalent  heat ;  and  for 
heat  of  considerable  intensity  —  is  small;  and  ty  being  small, 
t1 
^-  does  not  differ  much  from  — .     If  for  any  substance  t|t  were 
small  enough,  Sp  might  exceed  unity,  even  within  the  range  of 
heat  of  considerable  intensity,  in  which  case  there  would  be  dif- 
ferent values  of  p  for  different  wave-lengths.  The  molecular 
arrangements  of  such  a  substance  would  display  great  instability. 
A  molecule  formed  under  one  wave-length  would  be  decomposed 
and  otherwise  arranged  by  heat  of  a  different  kind.  As  we  have 
every  reason  to  believe  that  this  kind  of  instability  does  not  exist  in 
the  case  of  simple  substances,  we  may  infer  that  ^  is  not  very 
small,  the  general  conclusion  being  that  for  ordinary  substances 
ty  is  small  but  not  very  small. 
The  particular  value  of  i|r  which  for  any  substance  is  equal  to 
—  is  not  the  one  which  produces  the  greatest  effect  in  arranging 
&p 
the  atoms  into  groups.  The  greatest  effect  is  caused  by  the 
value  of  i/r  which  makes  (2^-f  l)i/r  equal  to  an  odd  multiple  of 
TT 
Z  ;  and  when  n  is  large,  such  a  value  must  exist.     In  this  case 
the  expression  for  the  relative  displacement  becomes  infinite. 
This  does  not  indicate  that  there  would  necessarily  be  a  rupture 
of  the  system,  but  merely  that  the  displacement  cannot  be  ex- 
pressed exactly  in  the  manner  supposed.  The  best  way  of  stating 
the  case  is,  that  a  vibration  of  that  exact  wave-length  cannot  be 
propagated  through  the  system  at  all.  By  the  effect  of  the 
factor  c,  by  which  in  the  second  approximation  it  becomes  ne- 
cessary to  multiply  jju,  the  time  of  vibration  and  the  wave-length 
are  increased ;  so  that  if  i/r  be  continuous,  this  particular  vibra- 
tion will  be  assimilated  to,  and  fall  in  with  that  correspond- 
ing to  some  smaller  value  of  yjr.  As  a  increases,  this  effect  in- 
creases and  extends  to  values  of  yfr  nearly  equal  (within  a  certain 
range)  to  the  particular  value  in  question. 
The  temperature  of  the  rth  atom  in  the  case  under  considera- 
tion is 
l*VX  co^(2n-2r+l)f 
4  cos2(2rc  +  l)^     ' 
X  being  some  constant ;  and  the  average  temperature  is 
fi*a*\,  y  cos2  (2tt-2r-f  l)ft  _  ^VX 
4n     *      cos2  (2n  +  l)^     ~  8  cos2  (2n+l)f  nearIy 
