﻿280 
  Mr. 
  H. 
  J. 
  E. 
  Beth 
  on 
  the 
  Oscillations 
  

  

  § 
  11. 
  n 
  r 
  -\- 
  n,; 
  + 
  n 
  z 
  — 
  n 
  u 
  = 
  p. 
  — 
  In 
  this 
  case 
  : 
  

  

  T 
  == 
  £i- 
  2 
  + 
  if 
  + 
  Ji* 
  + 
  1» 
  2 
  + 
  iP^ 
  2 
  + 
  Y„; 
  e 
  y 
  + 
  

  

  where 
  

  

  F 
  xx 
  = 
  ^a 
  xx 
  x 
  2 
  + 
  1 
  b 
  xx 
  if 
  + 
  ^ 
  x 
  .,~ 
  2 
  -f 
  ^d 
  xx 
  u 
  2 
  + 
  e^ 
  + 
  fzxPZ 
  

  

  + 
  g 
  xx 
  xu 
  + 
  /j^ys 
  + 
  k 
  xx 
  yu 
  + 
  Z^sw. 
  

   P^ 
  = 
  i«^a? 
  2 
  + 
  ^b 
  XI/ 
  y 
  2 
  -\-\c 
  nj 
  z 
  i 
  + 
  i^w 
  2 
  + 
  e 
  xy 
  xy 
  +f 
  xy 
  xz 
  

  

  + 
  g 
  xy 
  xu 
  + 
  Jht/yz 
  + 
  fc^yw 
  4- 
  ^sw. 
  

   Further 
  

  

  U 
  = 
  l«/x 
  2 
  + 
  \njf 
  + 
  in 
  z 
  h* 
  + 
  J«.V 
  + 
  U 
  4> 
  

  

  where 
  U 
  4 
  represents 
  a 
  homogeneous 
  function 
  or.' 
  order 
  four 
  

   in 
  .#, 
  y, 
  2 
  and 
  w. 
  

  

  The 
  result 
  is 
  that 
  the 
  disturbing 
  terms 
  of 
  the 
  second 
  kind 
  

   in 
  the 
  different 
  equations 
  are 
  the 
  derivatives, 
  resp. 
  according 
  

   to 
  x, 
  y, 
  z 
  and 
  u 
  of 
  : 
  

  

  ( 
  — 
  n 
  x 
  n 
  y 
  l 
  xy 
  T 
  n 
  x 
  n 
  z 
  k 
  xz 
  + 
  n 
  x 
  nji 
  xu 
  + 
  %/i^ 
  + 
  ftyWu/yu 
  ± 
  w 
  z 
  w 
  tt 
  ^ 
  —p)xyzu 
  

  

  (p 
  being 
  the 
  coefficient 
  of 
  a 
  term 
  xyzu 
  in 
  U 
  4 
  ) 
  . 
  

  

  Therefore 
  the 
  equations 
  of 
  motion 
  may 
  be 
  written 
  as 
  

   follows 
  : 
  

  

  - 
  , 
  2 
  d 
  R 
  n 
  

   # 
  + 
  7V#— 
  .~ 
  — 
  =0, 
  

  

  where, 
  if 
  — 
  / 
  is 
  written 
  instead 
  of 
  — 
  v 
  x 
  nj 
  xy 
  + 
  nnk 
  

  

  i 
  7 
  ~T~ 
  i 
  /' 
  J 
  J 
  X 
  Z 
  XZ 
  

  

  + 
  n 
  x 
  n 
  „h„ 
  + 
  n 
  v 
  n.y,jz 
  + 
  ii 
  y 
  n„t,j 
  U 
  ± 
  n 
  z 
  n 
  u 
  e 
  zu 
  — 
  p 
  : 
  

  

  R 
  = 
  K,.^ 
  + 
  K,y 
  + 
  + 
  M«^L 
  W 
  + 
  M.JBW 
  + 
  -/*y». 
  

  

  S, 
  = 
  2. 
  

   § 
  12. 
  The 
  only 
  relation 
  which 
  belongs 
  to 
  this 
  case 
  is 
  : 
  

  

  n 
  x 
  — 
  n 
  y 
  = 
  p 
  . 
  

  

  All 
  disturbing 
  terms 
  which 
  we 
  have 
  to 
  consider 
  are 
  of 
  

   order 
  h 
  3 
  . 
  Here 
  the 
  peculiarity 
  appears 
  that 
  all 
  disturbing 
  

   terms 
  of 
  the 
  second 
  kind 
  must 
  be 
  regarded 
  at 
  the 
  same 
  time 
  

   as 
  disturbing 
  terms 
  of 
  the 
  first 
  kind. 
  So 
  a 
  term 
  xy 
  1 
  gives 
  

  

  