﻿274 
  Mr. 
  H. 
  J. 
  E. 
  Beth 
  on 
  the 
  Oscillations 
  

  

  nearly 
  the 
  same 
  period 
  as 
  the 
  principal 
  vibration, 
  to 
  which 
  

   the 
  equation 
  in 
  which 
  the 
  term 
  indicated 
  appears, 
  relates 
  

   more 
  especially. 
  Such 
  terms, 
  as 
  is 
  known, 
  may 
  be 
  of 
  

   importance, 
  even 
  if 
  they 
  are 
  of 
  a 
  high 
  order 
  : 
  it 
  may 
  occur 
  

   that 
  they 
  are 
  of 
  influence 
  even 
  on 
  the 
  first 
  approximation. 
  

   In 
  order 
  to 
  determine 
  the 
  motion 
  at 
  first 
  approximation 
  it 
  

   will 
  be 
  necessary 
  to 
  include 
  such 
  terms 
  in 
  the 
  abridged 
  

   equations 
  of 
  motion. 
  

  

  Now 
  we 
  must 
  notice 
  that 
  terms 
  of 
  this 
  kind 
  appear 
  in 
  the 
  

   equations 
  of 
  motion 
  even 
  when 
  no 
  relation 
  exists. 
  The 
  term 
  

   xy 
  2 
  for 
  instance 
  will 
  after 
  substitution 
  as 
  above 
  give 
  rise 
  to 
  

   a 
  term 
  of 
  the 
  same 
  period 
  as 
  x. 
  That 
  these 
  terms 
  are 
  of 
  

   influence 
  on 
  the 
  first 
  approximation 
  we 
  see 
  from 
  the 
  deve- 
  

   lopments 
  (2, 
  p. 
  268). 
  The 
  modification 
  of 
  the 
  frequencies, 
  

   indicated 
  by 
  cr, 
  t, 
  , 
  . 
  . 
  is 
  the 
  effect 
  of 
  the 
  terms 
  w 
  T 
  e 
  refer 
  to. 
  

   These 
  terms 
  are 
  of 
  order 
  A 
  3 
  at 
  least. 
  

  

  We 
  shall 
  call 
  disturbing 
  terms 
  such 
  terms 
  of 
  higher 
  

   order 
  which 
  are 
  of 
  importance 
  for 
  the 
  first 
  approximation. 
  

   It 
  follows 
  from 
  what 
  is 
  said 
  above 
  that 
  we 
  have 
  to 
  

   distinguish 
  : 
  

  

  1st, 
  the 
  terms 
  which 
  are 
  always 
  disturbing, 
  even 
  when 
  no 
  

   relation 
  exists 
  ; 
  we 
  shall 
  call 
  them 
  disturbing 
  terms 
  of 
  the 
  

   first 
  kind. 
  

  

  2nd, 
  the 
  terms 
  which 
  owe 
  their 
  disturbing 
  property 
  to 
  the 
  

   existing 
  relation 
  ; 
  we 
  shall 
  call 
  them 
  disturbing 
  terms 
  of 
  the 
  

   second 
  kind. 
  

  

  In 
  a 
  single 
  case 
  (Si 
  = 
  2) 
  there 
  .appear 
  terms 
  which 
  are 
  

   disturbing 
  in 
  both 
  senses. 
  

  

  In 
  order 
  to 
  obtain 
  the 
  first 
  approximation 
  we 
  shall 
  admit 
  

   in 
  the 
  differential 
  equations 
  of 
  the 
  terms 
  of 
  higher 
  order 
  

   only 
  the 
  disturbing 
  ones, 
  and 
  of 
  these 
  terms 
  only 
  those 
  of 
  

   the 
  lowest 
  order. 
  We 
  shall 
  prove 
  that 
  in 
  each 
  of 
  the 
  cases 
  

   we 
  have 
  to 
  consider, 
  the 
  equations 
  of 
  motion 
  maybe 
  brought 
  

   into 
  this 
  form 
  : 
  

  

  x 
  + 
  nfx— 
  s 
  —0, 
  

   Ox 
  

  

  where 
  R 
  is 
  a 
  function 
  of 
  ,r, 
  ?/,.... 
  To 
  prove 
  this, 
  and 
  

   for 
  the 
  deduction 
  of 
  the 
  function 
  R, 
  we 
  shall 
  successively 
  

   discuss 
  the 
  cases 
  Si 
  = 
  3, 
  S 
  L 
  = 
  4, 
  and 
  Si 
  = 
  2. 
  

  

  