166         Prof.  M.  B.  Pell  on  the  Constitution  of  Matter. 
the  first  place,  that  probably  the  liquid  condition  cannot  exist 
with  any  permanence  except  under  the  combined  effects  of  tem- 
perature and  pressure ;  and  in  the  second  place,  I  must  anticipate 
so  much  as  to  say  that  I  hope  to  succeed  in  showing  that  it  is 
probable  that  the  atoms  of  a  solid,  under  the  action  of  heat,  ag- 
gregate themselves  into  molecules,  and  assume  the  liquid  and 
gaseous  conditions  at  a  far  lower  temperature  than  what  could 
correspond  to  the  velocity  necessary  to  carry  the  atom  from  A. 
to  B.  That  velocity  corresponds,  not  to  the  melting-point  of 
the  substance,  but  to  the  far  higher  temperature — higher  perhaps 
than  any  at  present  existing  in  the  solar  system,  under  which  a 
molecule  would  be  resolved  into  atoms. 
If  x  be  the  distance  between  the  atoms,  and/(a?)  the  dynamic 
cal  measure  of  the  attraction  between  them,  the  conditions  which 
have  been  stated  may  be  approximately  expressed  by  supposing 
/(*)  ="(«-«)  08-*)»^(*), 
where  OA=a,  OB  =  /3,  and  <p(x)  is  a  function  which  does  not 
change  sensibly  within  the  small  limits  <a?=a,  x=@.  Let 
/3— ct=h,  x=u+z,  h  and  z  being  supposed  small  compared 
with  a ;  then 
M=z(h-zf<l>(*  +  z) 
=z(h—z)s(j)(ct)  nearly. 
Put/'(a)=A3<£(a)=m2,  then 
/(*)=m^(l-?y 
It  must  be  observed  that  this  is  little,  if  any  thing,  more  than 
a  statement  in  a  mathematical  form  of  some  of  the  most  obvious 
properties  of  ordinary  matter.  It  remains  to  be  seen  whether 
this  statement  or  assumption  is  consistent  with,  and  can  explain 
other  and  more  recondite  properties. 
In  order  to  satisfy  approximately  the  condition  that  f'(/3) 
must  be  very  small,  I  have  made  it  zero,  giving  to  f(x)  =0 
three  roots  equal  to  ft.  Any  odd  number  of  roots  would  appa- 
rently do  as  well  as  three,  but  there  are  good  reasons  for  believing, 
as  I  shall  show  hereafter,  that  3  is  the  correct  index. 
The  equation  of  motion  is 
**z         ft  > 
which  may  be  solved  approximately  when  z  is  small  compared 
with  h.     Neglecting  the  last  term,  the  equation  may  be  written 
d*z  ,     9       3m2/  2     r>\ 
