Second  Proposition  of  the  Mechanical  Theory  of  Heat.     311 
least  action,  on  the  one  hand,  and  the  second  proposition  of  the 
theory  of  heat,  on  the  other,  I  considered  that  it  would  not  be 
uninteresting  to  investigate  the  question,  what  connexion  subsists 
between  the  second  proposition  of  the  theory  of  heat  and  Hamilton's 
dynamic  principle*  which  relates  to  varying  action. 
Hamilton's  principle  may  be  expressed  as  follows  f: — 
If  any  conservative  system  of  material  points  be  in  any  free 
motion  between  any  initial  and  final  configuration,  the  following 
equation  will  be  universally  valid  for  an  infinitely  small  altera- 
tion of  this  motion  : — 
BA  =  ^mvlSsi—^mv0Ss0-\'iSE>     .     .     .     .     (1) 
where  m  is  the  mass  of  a  point  of  the  system ;  8sl  and  Ss0  are 
the  displacements  of  the  same  point  from  the  previous  final  con- 
figuration into  the  new  one,  and  from  the  previous  initial  confi- 
guration into  the  new  one,  respectively;  vL  and  v0  denote  the 
velocity^  measured  always  in  the  direction  of  displacement,  of  the 
same  point  in  the  earlier  final  configuration  and  initial  configu- 
ration respectively ;  i  is  the  time  during  which  the  system  passes 
from  the  first  initial  configuration  to  the  first  final  configuration. 
8 A.  is  the  difference  of  action,  and  8E  the  difference  of  total 
energy,  between  the  new  and  the  former  path.  By  action  is 
understood  the  time-integral  of  twice  the  vis  viva  for  the  time 
during  which  the  system  passes  from  the  initial  to  the  final  con- 
figuration ;  by  total  energy,  the  sum  of  the  kinetic  and  potential 
energies  present  at  a  determined  moment.  Thus  both  A  andE 
in  one  and  the  same  path  are  constant,  in  whatever  configura- 
tion the  system  may  be,  but  in  general  variable  from  path  to  path. 
If  the  total  vis  viva  of  the  system  at  a  fixed  time  be  T,  then 
f 
2T.  dt.     ......     (2) 
If,  further,  U  be  the  potential  energy  of  the  system  at  the  same 
instant,  then  the  total  energy 
E  =  T  +  U. 
Both  T  and  U  have  different  values  at  different  points  of  the 
path  ;  but  their  sum  is  the  same  constant  quantity  for  all  points 
of  the  path.     Hence 
i.E=ri(T  +  U)^ (3) 
Jo 
Taking  into  consideration  equations  (2)  and  (3),  Hamilton's 
*  Hamilton  :  "  On  a  General  Method  in  Dynamics,"  Phil.  Trans.  1.834  ; 
"  Second  Essay  on  a  General  Method  in  Dynamics/'  ibid.  1835. 
f  Conf.  Sir  W.  Thomson  and  P.  G.  Tait,  'Treatise  on  Natural  Philo- 
sophy/ vol.  i.  p.  235. 
