﻿276 
  Mr. 
  H. 
  J. 
  E. 
  Beth 
  on 
  the 
  Oscillations 
  

  

  manner, 
  that 
  A, 
  B, 
  X, 
  and 
  fi 
  are 
  o£ 
  order 
  h 
  or 
  smaller. 
  Then 
  

   we 
  may 
  replace 
  in 
  the 
  second 
  member 
  o£ 
  the 
  equations 
  : 
  

  

  x 
  2 
  by 
  n,»(AW-*»), 
  f 
  by 
  n 
  y 
  \Wh 
  2 
  -f), 
  

  

  x 
  „ 
  — 
  w/a?, 
  ,y 
  „ 
  — 
  n 
  2 
  y. 
  

  

  If 
  we 
  take 
  this 
  into 
  account 
  for 
  the 
  disturbing 
  terms 
  and 
  

   if 
  we 
  omit 
  the 
  non-disturbing 
  terms, 
  then 
  the 
  equations 
  

   become 
  : 
  

  

  C 
  x 
  + 
  n 
  2 
  x 
  = 
  (bn 
  2 
  + 
  en 
  2 
  + 
  2p)xy 
  — 
  bivy, 
  

  

  (The 
  terms 
  2pxy 
  in 
  the 
  first 
  equation 
  and 
  px 
  2 
  in 
  the 
  second 
  

   originate 
  from 
  a 
  term 
  —px 
  2 
  y, 
  appearing 
  in 
  U 
  3 
  .) 
  

  

  To 
  get 
  rid 
  of 
  the 
  term 
  with 
  xy 
  we 
  notice 
  that 
  icy 
  by 
  

   substitution 
  of 
  the 
  expressions 
  for 
  x 
  and 
  y 
  gives 
  rise 
  to 
  the 
  

   same 
  disturbing 
  term 
  as 
  n 
  x 
  n 
  xy. 
  So 
  we 
  may 
  replace 
  —b'xy 
  

   by 
  -bn 
  x 
  n 
  y 
  xy. 
  ^ 
  

  

  We 
  replace 
  in 
  the 
  terms 
  of 
  higher 
  order 
  n 
  by 
  2n 
  x 
  . 
  Then 
  

   the 
  equations 
  become 
  : 
  

  

  C 
  x 
  + 
  n 
  x 
  2 
  x= 
  (±cn 
  2 
  — 
  hn 
  x 
  - 
  + 
  2p)xy, 
  

  

  \ 
  y 
  + 
  v^ 
  = 
  ( 
  2c7 
  V* 
  ~ 
  iK 
  2 
  + 
  p> 
  2 
  - 
  

  

  We 
  may 
  write 
  the 
  equations 
  in 
  the 
  form 
  : 
  

  

  # 
  + 
  w 
  r 
  # 
  — 
  -^ 
  — 
  =U, 
  

   .J 
  - 
  d^' 
  

  

  where, 
  if 
  we 
  write 
  — 
  d 
  2 
  for 
  2en*—\ibn*-\rp 
  ' 
  

   R 
  = 
  — 
  d 
  2 
  xh/. 
  

  

  § 
  7. 
  n~, 
  + 
  r) 
  y 
  — 
  n 
  z 
  = 
  p. 
  In 
  this 
  case 
  

  

  u=kv 
  + 
  iw 
  y 
  y 
  + 
  \n 
  2 
  z 
  2 
  + 
  u 
  3 
  , 
  

  

  where 
  U 
  3 
  is 
  a 
  homogeneous 
  function 
  of 
  order 
  three 
  in 
  

   #, 
  y, 
  and 
  ~. 
  

  

  where 
  e. 
  g. 
  

  

  The 
  equations 
  of 
  Lagrange 
  can 
  be 
  written 
  down 
  again. 
  

   When 
  in 
  the 
  terms 
  of 
  the 
  2nd 
  order 
  x, 
  y, 
  and 
  z 
  are 
  replaced 
  

   resp. 
  by 
  —n 
  2 
  x 
  } 
  — 
  n 
  2 
  y, 
  and 
  — 
  n 
  z 
  2 
  z, 
  then 
  we 
  have 
  e. 
  g. 
  in 
  the 
  

  

  