﻿about 
  a 
  Position 
  of 
  Equilibrium. 
  275 
  

  

  Si 
  = 
  3. 
  

   § 
  5. 
  There 
  are 
  two 
  relations 
  to 
  bo 
  considered, 
  namely 
  : 
  

   1°, 
  2n 
  -n 
  =p, 
  

  

  X 
  // 
  r 
  ' 
  

  

  2°, 
  n 
  x 
  +n 
  y 
  -n=p. 
  

  

  When 
  one 
  of 
  those 
  relations 
  exists 
  there 
  are 
  in 
  the 
  equations 
  

   of 
  motion 
  disturbing 
  terms 
  of 
  the 
  second 
  kind 
  of 
  order 
  Jr. 
  

   ( 
  . 
  </. 
  the 
  term 
  with 
  xy 
  in 
  the 
  first 
  equation 
  in 
  case 
  of 
  the 
  first 
  

   relation. 
  So 
  the 
  disturbing 
  terms 
  of 
  the 
  first 
  kind 
  can 
  be 
  

   left 
  out, 
  because 
  they 
  are 
  of 
  order 
  h 
  z 
  at 
  least, 
  and 
  because 
  

   we 
  admit 
  the 
  disturbing 
  terms 
  of 
  the 
  lowest 
  order 
  only. 
  

  

  It 
  is 
  clear 
  that 
  the 
  disturbing 
  terms 
  of 
  the 
  second 
  kind 
  of 
  

   order 
  Ir 
  appear 
  only 
  in 
  the 
  equations 
  resp. 
  for 
  as 
  and 
  //, 
  and 
  

   for 
  ./•. 
  v, 
  and 
  c. 
  They 
  contain 
  no 
  other 
  but 
  these 
  coordinates. 
  

   So 
  we 
  may 
  say 
  : 
  

  

  /// 
  the 
  case 
  Si 
  =.3 
  the 
  coordinates 
  which 
  appear 
  in 
  the 
  rela- 
  

   tion 
  behave 
  at 
  first 
  approximation 
  as 
  if 
  they 
  were 
  the 
  only 
  

   coordinates; 
  the 
  remaining 
  coordinates 
  behave 
  at 
  first 
  approxi- 
  

   mation 
  as 
  if 
  no 
  relation 
  existed. 
  

  

  It 
  will 
  therefore 
  be 
  sufficient 
  to 
  consider 
  a 
  mechanism 
  

   which 
  has 
  2, 
  resp. 
  3 
  degrees 
  of 
  freedom. 
  

  

  § 
  G. 
  2n 
  r 
  —n 
  1/ 
  = 
  p. 
  As 
  Ave 
  have 
  to 
  include 
  in 
  the 
  differential 
  

  

  equations 
  no 
  terms 
  of 
  a 
  higher 
  order 
  than 
  Ji 
  2 
  , 
  we 
  have 
  in 
  the 
  

   expressions 
  for 
  the 
  kinetic 
  energy 
  T 
  and 
  for 
  the 
  potential 
  

   energy 
  U 
  to 
  take 
  no 
  terms 
  of 
  a 
  higher 
  order 
  than 
  /r 
  3 
  . 
  So 
  

   we 
  may 
  write 
  : 
  

  

  " 
  T 
  = 
  W 
  + 
  if 
  + 
  T, 
  : 
  D 
  = 
  I 
  n 
  x 
  V 
  + 
  h;hf 
  + 
  U 
  s 
  , 
  

  

  where 
  U 
  3 
  is 
  a 
  homogeneous 
  function 
  of 
  order 
  three 
  in 
  x, 
  //, 
  

   and 
  

  

  T 
  3 
  = 
  Ua.rP 
  + 
  by'x 
  2 
  + 
  2cx'scy 
  + 
  2dyxy 
  + 
  exy 
  2 
  +-fyf). 
  

  

  The 
  equations 
  of 
  Lagrange 
  become 
  : 
  

   J 
  'x 
  + 
  n 
  fx— 
  —\a,'r— 
  axx—by'x—bxy—-cxy--dyy 
  + 
  (le 
  — 
  d)y 
  2 
  — 
  -^ 
  - 
  . 
  

  

  \^y+n 
  y 
  2 
  y=(^—c)P—cxx—dyw—exy—exy—fyy—ify 
  2 
  -- 
  * 
  

  

  The 
  disturbing 
  terms 
  are 
  : 
  

  

  in 
  the 
  firsi 
  equation 
  those 
  with 
  yx, 
  xy, 
  xy, 
  xy 
  : 
  

  

  second 
  „ 
  „ 
  „ 
  .*/■-, 
  xx\ 
  n". 
  

  

  N 
  >w 
  we 
  shall 
  try 
  to 
  satisfy 
  the 
  equations 
  in 
  first 
  approxi- 
  

   mation 
  by 
  

  

  ,r 
  = 
  A/i 
  cos 
  f// 
  / 
  + 
  \), 
  // 
  = 
  B/' 
  cos 
  (n 
  J 
  + 
  fi), 
  

   where 
  A. 
  B, 
  \. 
  and 
  fi 
  are 
  functions 
  of 
  /. 
  however 
  in 
  such 
  a 
  

  

  