﻿about 
  a 
  Position 
  of 
  Equilibrium. 
  281 
  

  

  ri?e 
  to 
  a 
  term 
  with 
  cos(n. 
  r 
  t 
  + 
  \) 
  and 
  a 
  term 
  with 
  

  

  eos{(2«y— 
  n 
  x 
  )t+2fi-\} 
  } 
  

   which 
  l>oth 
  are 
  disturbing 
  in 
  the 
  conation 
  for 
  x. 
  

  

  The 
  disturbing 
  terms 
  of 
  the 
  first 
  kind 
  are 
  the 
  same 
  as 
  

   for 
  S,=4. 
  

  

  The 
  disturbing 
  terms 
  of 
  the 
  second 
  kind 
  contain 
  no 
  other 
  

   coordinates 
  but 
  x 
  and 
  y 
  and 
  they 
  appear 
  only 
  in 
  the 
  equations 
  

  

  for 
  x 
  and 
  y. 
  Therefore 
  we 
  may 
  for 
  the 
  discussion 
  of 
  these 
  

   term- 
  restrict 
  ourselves 
  to 
  a 
  mechanism 
  of 
  two 
  degrees 
  of 
  

   freedom. 
  

  

  T 
  and 
  U 
  have 
  the 
  same 
  form 
  as 
  in 
  the 
  case 
  3n 
  x 
  — 
  n 
  y 
  = 
  p 
  

   (p. 
  278). 
  

  

  The 
  result 
  is 
  that 
  the 
  disturbing 
  terms 
  of 
  the 
  second 
  kind 
  

   in 
  the 
  two 
  equations 
  are 
  the 
  derivatives 
  resp. 
  according 
  to 
  

   X 
  and 
  y 
  of 
  : 
  

  

  «x 
  8 
  { 
  i 
  (} 
  2 
  &zz 
  + 
  a^X 
  Z 
  lJ 
  + 
  K 
  - 
  C 
  x 
  , 
  + 
  llsy 
  — 
  dyy) 
  *Y 
  + 
  3 
  (i^W 
  + 
  C 
  *y) 
  3Cy 
  Z 
  } 
  + 
  

  

  — 
  e- 
  2 
  oc 
  z 
  y 
  — 
  e 
  3 
  x 
  2 
  y* 
  — 
  e 
  v 
  ry 
  3 
  

  

  (e,. 
  e-i, 
  and 
  e± 
  being 
  the 
  coefficients 
  of 
  the 
  terms 
  with 
  x 
  z 
  y, 
  x 
  2 
  y 
  2 
  , 
  

   and 
  xy 
  3 
  , 
  appearing 
  in 
  U 
  4 
  ). 
  

  

  The 
  equations 
  of 
  motion 
  may 
  be 
  written 
  in 
  the 
  following 
  

   form 
  : 
  

  

  f.. 
  2 
  BR 
  n 
  

   x+n?x— 
  ^— 
  =0, 
  

  

  J 
  °' 
  v 
  

  

  I 
  •• 
  i 
  2 
  BR 
  n 
  . 
  

  

  where, 
  if 
  

  

  e 
  2 
  ' 
  = 
  e 
  , 
  — 
  InjfQJ'st 
  + 
  a 
  X!/ 
  ), 
  

  

  ez=e 
  z 
  + 
  $n 
  x 
  2 
  (c 
  xx 
  — 
  2b 
  ^ 
  + 
  a^), 
  

  

  el 
  = 
  ei 
  — 
  ^n/Qb^ 
  + 
  c 
  xl/ 
  ). 
  

  

  R 
  = 
  K^ 
  + 
  K, 
  .//* 
  + 
  U 
  xx 
  A 
  2 
  h 
  2 
  x' 
  2 
  + 
  - 
  e,'x 
  3 
  y 
  - 
  e 
  5 
  'x*f 
  - 
  elxy\ 
  

  

  \ 
  13. 
  We 
  have 
  now 
  brought 
  the 
  equations 
  of 
  motion 
  for 
  

   the 
  different 
  cases 
  we 
  have 
  to 
  consider 
  in 
  this 
  form 
  : 
  

  

  x 
  + 
  n 
  x 
  ~x 
  — 
  =— 
  - 
  = 
  0, 
  I 
  

  

  - 
  . 
  o 
  BR 
  ft 
  L 
  (6) 
  

  

  where 
  R 
  is 
  a 
  function 
  of 
  the 
  coordinates 
  x 
  3 
  y 
  : 
  the 
  form 
  

  

  of 
  R 
  we 
  have 
  deduced 
  for 
  the 
  different 
  cas< 
  

  

  Phil. 
  Mag. 
  S. 
  6. 
  Vol. 
  26. 
  No. 
  152. 
  Aug. 
  1913. 
  U 
  

  

  