340  M.  C.  Szily  on  Hamilton's  Principle  and  the 
The  first  who,  to  my  knowledge,  occupied  himself  with  this 
question  was  Ludwig  Bolzmann.  His  memoir  relative  to  it 
was  presented  to  the  Vienna  Academy  on  the  8th  of  February, 
1866,  and  appeared  in  the  Sitzungsberichte  under  the  title  "  On 
the  Mechanical  Signification  of  the  Second  Proposition  of  the 
Theory  of  Heat." 
Independently  of  Bolzmann,  and  evidently  unaware  of  the 
existence  of  his  memoir,  Clausius  laid  before  the  Society  for 
Natural  and  Medical  Science  of  the  Lower  Rhine,  on  the  7th 
November  1870,  a  memoir  entitled  "  On  the  Reduction  of  the 
Second  Axiom  of  the  Mechanical  Theory  of  Heat  to  general 
Mechanical  Principles"*. 
The  result  in  both  memoirs  is  much  the  same  : — "  The  second 
proposition  of  the  mechanical  theory  of  heat  is  capable  of  being 
explained  from  the  principles  of  analytical  mechanics.  For  this 
purpose,  however,  new  and  peculiar  developments  are  necessary ; 
and  the  calculations  relative  to  it  are  very  similar  to  those  ge- 
nerally made  use  of  in  order  to  demonstrate  the  so-called  f  prin- 
ciple of  least  action/  " 
Clausius  investigated,  in  the  first  place,  what  connexion  exists 
between  the  periodical  motions  of  a  material  point  in  a  closed 
path,  presupposing  conservative  forces  (that  is,  possessing  a 
force-function) ;  he  next  discusses  the  possible  causes  of  altera- 
tion of  path,  and  shows  the  validity  of  the  equations  advanced 
in  the  different  cases.  In  the  third  part  of  his  memoir  Clausius 
passes  from  this  simple  case  to  more  complicated  ones,  by  assu- 
ming a  whole  system  of  reciprocally  acting  material  points  in 
periodical  motion  in  closed  paths.  He  then  generalizes  the 
equations  for  such  stationary  motions  as  do  not  take  place  in 
closed  paths.  The  mechanical  equation  thus  deduced,  and  with 
which  he  next  compares  the  second  proposition  of  the  theory 
of  heat,  is  the  following : — 
SL=Sf+2TSlog>", 
in  which  SL  denotes  the  work  which  the  conservative  forces 
must  accomplish  that  the  system  may  pass  from  a  given  station- 
ary motion  into  another  stationary  motion ; 
signifies  the  meanwhile  resulting  variation  of  the  mean  vis  viva 
of  the  system,  and  %  the  duration  of  a  motion-period. 
My  attention  having  been  directed  by  Clausius's  memoir  to 
the  relation  subsisting  between  his  equation  and  the  principle  of 
*  Phil.  Mag.  S.  4.  vol.  xlii.  p.  161. 
