Prof.  M.  B.  Pell  on  the  Constitution  of  Matter.         177 
to  be  nearly  equal  to  some  odd  multiple  of  ~,  so  that  cos  {2n  +  l)yfr 
is  small,  but  a  so  small  that  the  character  of  the  vibrations  is 
maintained.     Since 
sin  (2>i-2r  +  2)^=cos  (2r— l)f  nearly, 
the  relative  displacement  is  a  maximum  and  large  compared  with 
a  for  such  values  of  ras  make  (2r— •  l)^r  a  multiple  of  7r  or  most 
nearly  so;  and  this,  of  course,  occurs  at  regular  intervals.  If;; 
be  the  whole  number  most  nearly  satisfying  the  condition  2pyfr  = 
it,  the  atoms  are  arranged  in  groups  of  p  each,  where  p  is  a  num- 
ber depending  upon  the  wave-length  of  the  disturbance  and  the 
nature  of  the  system,  and  not  at  all  upon  a  or  the  intensity  of 
the  disturbance.  Suppose  now  a  or  the  temperature  to  increase 
gradually ;  the  groups  remain  the  same,  but  become  more  and 
more  isolated.  Each  group  acquires  an  oscillatory  motion  as  a 
whole  in  addition  to  the  vibratory  motion  of  its  atoms  amongst 
themselves;  and  the  time  of  this  oscillation  corresponds  nearly 
to  the  fundamental  or  lowest  note  of  a  group  of  p  atoms  vibra- 
ting without  restraint ;  for  in  that  case 
^1==2msin —  =  2m sin ^jr=fi  nearly. 
The  disturbance  a  cos  fit  representing  the  prevailing  heat,  fi  and 
therefore  yjr  also,  for  atoms  of  a  given  kind,  has  a  certain  very 
limited  range  of  value.    We  may  suppose,  then,  that  there  is 
77* 
some  one  particular  value  of  yjr  making   — -=p  an  integer;  and 
that  particular  wave-length  corresponds  exactly  to  the  funda- 
mental note  of  the  group,  or  molecule,  of  p  atoms. 
If  we  suppose  now  that  a  increases  until  the  maximum  value 
of  the  relative  displacement  exceeds  a  certain  quantity,  the  se- 
verance becomes  complete.  This  does  not  necessarily  take  place 
at  all  points  at  once;  for  the  weakening  of  the  connexion  between 
the  groups  would  impede  the  propagation  of  the  disturbance. 
The  first  group  would  first  melt  off,  or,  if  the  temperature  were 
higher,  would  fly  off  as  a  molecule  of  vapour;  and  the  next 
group  would  then  be  directly  exposed  to  the  disturbance,  and  be 
melted  or  evaporated  in  its  turn.  It  should  be  observed  that 
the  energy  employed  in  severing  the  connexion  between  the 
groups  does  not  increase  the  temperature,  but  becomes  latent. 
I  must  remark  here  that  the  liquid  condition  would  be  better 
accounted  for,  and  some  other  phenomena  perhaps  explained,  by 
supposing  the  three  roots  equal  to  /3  of  the  equation  f{pc)=0  to 
be  replaced  by  three  roots  nearly  equal  to  each  other — that  is, 
roots  whose  differences  are  generally  small  compared  with  h.     If 
Phil.  Mag.  S.  4.  Vol.  43.  No.  285.  March  1872.  N 
