﻿about 
  a 
  Position 
  of 
  Equilibrium. 
  283 
  

  

  then 
  by 
  substitution 
  of 
  the 
  expressions 
  found 
  for 
  .i\ 
  //, 
  in 
  

  

  R 
  this 
  R 
  will 
  become 
  a 
  function 
  of 
  the 
  a's, 
  the 
  /S's, 
  and 
  t. 
  

   The 
  variability 
  of 
  the 
  as 
  and 
  /Q's 
  with 
  the 
  time 
  is 
  then 
  

   given 
  by 
  

  

  du. 
  ^b\\ 
  d/3s_ 
  _BR 
  

  

  dt 
  B/3,' 
  dt 
  ~ 
  -du/ 
  

  

  dt 
  

  

  BR 
  

  

  b/v 
  

  

  

  BR 
  

  

  B*y 
  

  

  (7) 
  

  

  § 
  15. 
  If 
  now 
  we 
  solve 
  the 
  abridged 
  equations 
  

  

  arising 
  from 
  (6) 
  (p. 
  281) 
  by 
  omission 
  iof 
  the 
  terms 
  ^ 
  

  

  BR 
  

  

  X 
  = 
  

  

  according 
  to 
  the 
  method 
  of 
  Hamilton-Jacobi, 
  we 
  

   may 
  arrive 
  at 
  

  

  ^ 
  x 
  cos(n 
  3 
  t+2n 
  x 
  p 
  x 
  ),"} 
  

  

  i 
  

   /y= 
  ^cos(V 
  + 
  2^), 
  f 
  " 
  ' 
  ' 
  • 
  (8) 
  

  

  J 
  

  

  We 
  must 
  suppose 
  a 
  r 
  , 
  a 
  s 
  , 
  to 
  be 
  of 
  order 
  A 
  2 
  , 
  as 
  the 
  

  

  -amplitudes 
  of 
  the 
  vibrations 
  must 
  be 
  of 
  order 
  h. 
  

  

  When 
  we 
  perform 
  the 
  substitution 
  of 
  the 
  expressions 
  (8) 
  

   in 
  R 
  we 
  meet 
  with 
  two 
  sorts 
  of 
  terms, 
  namely, 
  those 
  which 
  

   contain 
  t 
  explicitly 
  and 
  those 
  which 
  do 
  not 
  contain 
  t 
  explicitly. 
  

   Onlv 
  the 
  terms 
  of 
  this 
  last 
  sort 
  are 
  of 
  importance 
  for 
  the 
  

   tirst 
  approximation 
  ; 
  the 
  others 
  we 
  omit. 
  

  

  From 
  a 
  term 
  xfifz'nJ 
  1 
  arises 
  by 
  the 
  substitution 
  only 
  one 
  

   term 
  with 
  cosine, 
  which 
  does 
  not 
  contain 
  t 
  (by 
  y/, 
  y, 
  r, 
  ~s 
  are 
  

   meant 
  the 
  absolute 
  values 
  of 
  the 
  coefficients 
  p, 
  q, 
  r, 
  s 
  of 
  

   n 
  y 
  /> 
  ,, 
  n 
  ., 
  n 
  u 
  in 
  the 
  relation). 
  This 
  term 
  is 
  : 
  

  

  1 
  

  

  •?S— 
  1 
  p 
  ij 
  r 
  

  

  where 
  S=p 
  + 
  q+ 
  r+ 
  s. 
  

  

  The 
  term 
  pii 
  c 
  'v 
  2 
  appearing 
  in 
  R 
  (§ 
  13, 
  p. 
  281) 
  gives 
  rise 
  

   p 
  

   to 
  a 
  term 
  ~—,*v> 
  

   - 
  

  

  os 
  2 
  ( 
  imS, 
  + 
  M 
  + 
  rn 
  fiz 
  + 
  sn*&*) 
  , 
  

  

  D 
  2 
  

  

  