﻿272 
  Mr. 
  H. 
  J. 
  E. 
  Beth 
  on 
  the 
  Oscillat 
  

  

  ions 
  

  

  appear 
  among 
  other 
  terms 
  those 
  of 
  the 
  first 
  degree 
  and 
  

   second 
  kind, 
  which 
  have 
  risen 
  4 
  orders. 
  So 
  this 
  term 
  

   rises 
  2 
  + 
  4 
  = 
  6 
  orders. 
  The 
  P 
  function 
  for 
  the 
  term 
  of 
  second 
  

   degree 
  and 
  second 
  kind 
  rises 
  6 
  orders 
  also 
  ; 
  so 
  the 
  term 
  

   rises 
  2 
  + 
  6 
  = 
  8 
  orders. 
  Both 
  terms 
  of 
  the 
  second 
  degree 
  rise 
  

   to 
  the 
  same 
  order. 
  

  

  In 
  this 
  way 
  we 
  may 
  go 
  on. 
  It 
  will 
  be 
  clear 
  that 
  both 
  

   terms 
  of 
  the 
  same 
  degree 
  rise 
  to 
  the 
  same 
  order. 
  Each 
  

   following 
  term 
  rises 
  4 
  orders 
  more 
  than 
  the 
  former. 
  The 
  

   order 
  of 
  the 
  term 
  of 
  the 
  first 
  kind 
  and 
  Mi 
  degree 
  being 
  in 
  

   general 
  1$> 
  X 
  — 
  1, 
  the 
  order 
  of 
  both 
  terms 
  of 
  the 
  /cth 
  degree 
  

   will 
  become 
  • 
  

  

  £S 
  1 
  _l_(£_l)4_2 
  = 
  £(S 
  1 
  --4) 
  + 
  l. 
  

  

  § 
  3. 
  From 
  this 
  result 
  it 
  follows 
  immediately 
  that 
  there 
  

   exists 
  a 
  considerable 
  difference 
  between 
  the 
  two 
  cases 
  Sx<4 
  

  

  and 
  Si>4. 
  In 
  the 
  latter 
  case 
  the 
  vibrations 
  of 
  relation 
  

   remain 
  feeble 
  in 
  respect 
  to 
  the 
  principal 
  vibrations 
  ; 
  the 
  

   intensity 
  is 
  smaller 
  for 
  the 
  vibrations 
  of 
  a 
  higher 
  degree. 
  

  

  In 
  the 
  case 
  S 
  x 
  = 
  4, 
  however, 
  all 
  the 
  vibrations 
  of 
  relation 
  

   reach 
  the 
  order 
  of 
  greatness 
  of 
  the 
  principal 
  vibrations. 
  In 
  

   the 
  cases 
  Si 
  = 
  3 
  and 
  Si 
  = 
  2, 
  it 
  should 
  seem 
  as 
  if 
  the 
  vibra- 
  

   tions 
  of 
  relation 
  would 
  have 
  an 
  intensity 
  which 
  is 
  great 
  in 
  

   respect 
  to 
  the 
  intensity 
  of 
  the 
  principal 
  vibrations. 
  This 
  

   conclusion 
  is 
  not 
  exact, 
  as 
  Prof. 
  Korteweg 
  shows 
  by 
  a 
  special 
  

   investigation 
  into 
  these 
  cases. 
  However, 
  it 
  will 
  be 
  clear 
  

   that 
  the 
  development 
  in 
  series 
  (2) 
  have 
  lost 
  their 
  validity 
  

   completely 
  as 
  soon 
  as 
  the 
  intensity 
  of 
  the 
  motion 
  has 
  become 
  

   so 
  great 
  that 
  the 
  vibrations 
  of 
  relation 
  reach 
  the 
  intensity 
  of 
  

   the 
  principal 
  vibrations. 
  

  

  Therefore 
  in 
  the 
  cases 
  S 
  2 
  <C 
  4, 
  as 
  soon 
  as 
  the 
  motion 
  of 
  the 
  

  

  mechanism 
  has 
  a 
  certain 
  intensity 
  the 
  developments 
  in 
  series 
  

   can 
  no 
  longer 
  represent 
  the 
  motion. 
  In 
  what 
  follows 
  we 
  

   shall 
  investigate 
  what 
  becomes 
  of 
  the 
  motion 
  in 
  the 
  case 
  

   mentioned. 
  

  

  In 
  Prof. 
  Korteweg's 
  paper 
  it 
  is 
  shown 
  that 
  the 
  develop- 
  

   ments 
  lose 
  their 
  validity 
  in 
  the 
  case 
  Si 
  = 
  3 
  as 
  soon 
  as 
  - 
  is 
  of 
  

  

  the 
  same 
  order 
  of 
  greatness 
  as 
  — 
  ; 
  in 
  the 
  cases 
  Si 
  = 
  4 
  and 
  

  

  (h\ 
  2 
  „ 
  Ux 
  

  

  S 
  x 
  = 
  2 
  as 
  soon 
  as 
  I 
  =- 
  1 
  is 
  of 
  the 
  same 
  order 
  of 
  greatness 
  as 
  

  

  £-. 
  Therefore 
  we 
  shall 
  suppose 
  — 
  to 
  be 
  of 
  order 
  - 
  in 
  the 
  

  

  n 
  x 
  n 
  x 
  I 
  

  

  case 
  S 
  : 
  = 
  3, 
  and 
  of 
  order 
  lj) 
  in 
  the 
  cases 
  Sx 
  = 
  4 
  and 
  

  

  