342         M.  C.  Szily  on  the  Mechanical  Theory  of  Heat, 
principle,  expressed  in  equation  (1),  assumes  the  following  more 
familiar  form : — 
bC (T-V)dt=^mvl8sl-Xmv0Ss0-^Bi. . . . 
Returning  to  the  first  form,  let  us  inquire  when  is  the  varia- 
tion of  the  action  independent  of  the  initial  and  final  configura- 
tions ?     This  will  be  the  case  when 
^mv^s^^mvQBsQj — (4) 
that  is,  when  the  action  in  the  time  during  which  the  system  is 
passing  from  the  previous  to  the  new  initial  configuration  is  just 
equal  to  the  action  during  the  passage  of  the  system  from  the 
previous  to  the  new  final  configuration. 
The  condition  in  (4)  is,  e.  g.,  satisfied  : — 
When  the  paths  all  start  from  a  common  initial  position  and 
pass  over  to  a  common  final  position;  for  then  Bs1=-0 
and  Ss0=0  for  every  point  of  the  system  ;  or 
When  the  paths  are  closed  and  the  motions  periodic ;  for 
then  Ssl  =  Bs0  and  vl=v0  for  every  point  of  the  system ;  or 
When  the  paths  are  not  closed,  but  the  displacement  of 
every  point  in  the  initial  and  in  the  final  configuration 
proceeds  according  to  the  equation  v1Bs1  =  vQ8s0. 
All  these  are  only  special  cases ;  the  general  condition  is  given 
in  equation  (4). 
When,  in  the  motion  of  the  system,  the  alteration  of  the  mo- 
tion satisfies  equation  (4),  Hamilton's  principle  can  be  expressed 
very  simply :. — 
8A=t8E; (5) 
that  is,  the  variation  of  the  action  in  the  transition  from  one  path 
to  another  is  equal  to  the  product  of  the  time  necessary  for  the 
accomplishment  of  the  path,  into  the  variation  of  the  total  energy. 
Let  T  be  the  mean  value  of  the  total  vis  viva  during  a  period 
of  the  motion ;  then 
~  =  (lTdt, 
H: 
and 
A=2iT*, 
*  Introducing  this  value  of  A  into  equation  (5),  and  remembering  that, 
according  to  equation  (3), 
E  =  f+U, 
it  follows  that 
2OT+2Tfo'=iST-HSU, 
and  hence 
SU=ST+2T51og«; 
and  this  is  the  equation  of  Clausius. 
