﻿about 
  a 
  Position 
  of 
  Equilibrium. 
  271 
  

  

  The 
  vibrations 
  of 
  relation 
  of: 
  the 
  second 
  kind 
  correspond 
  to 
  

  

  p=*Pi 
  + 
  lj 
  q=kqi 
  

  

  The 
  vibrations 
  of 
  relation, 
  which 
  correspond 
  to 
  different 
  

   values 
  of 
  /•. 
  are 
  called 
  of 
  different 
  degree. 
  

  

  The 
  frequencies 
  of 
  the 
  vibrations 
  of 
  relation 
  of 
  the 
  first 
  

   and 
  second 
  kind 
  and 
  of 
  the 
  Mi 
  degree 
  are 
  resp. 
  n 
  x 
  + 
  kp 
  + 
  ]&h? 
  

  

  and 
  /? 
  . 
  — 
  /p 
  — 
  E/r, 
  where 
  E 
  depends 
  on 
  the 
  values 
  of 
  A, 
  B, 
  etc. 
  

  

  What 
  now 
  is 
  found 
  for 
  the 
  coordinate 
  ,r 
  is 
  also 
  exact 
  for 
  

   // 
  and 
  for 
  all 
  the 
  coordinates 
  which 
  enter 
  in 
  the 
  relation 
  (5). 
  

   So 
  we 
  may 
  say 
  : 
  

  

  When 
  there 
  exists 
  a 
  linear 
  relation 
  of 
  the 
  form 
  (5) 
  then 
  

   there 
  are 
  a 
  number 
  of 
  vibrations 
  of 
  higher 
  order, 
  which 
  

   assume 
  an 
  abnormally 
  great 
  intensity 
  ; 
  their 
  frequencies 
  lie 
  

   in 
  the 
  vicinity 
  of 
  and 
  on 
  both 
  sides 
  of 
  the 
  principal 
  fre- 
  

   quencies 
  which 
  appear 
  in 
  the 
  relation. 
  

  

  §2. 
  Prof. 
  Korteweg 
  now 
  investigates 
  to 
  which 
  order 
  of 
  

   greatness 
  the 
  vibrations 
  of 
  relation 
  will 
  rise 
  in 
  general. 
  

   This 
  depends 
  on 
  the 
  intensity 
  of 
  the 
  motion 
  of 
  the 
  mechanism, 
  

   i.e. 
  on 
  It. 
  f 
  takes 
  the 
  value 
  pQ 
  + 
  D/r. 
  If 
  now 
  the 
  in- 
  

  

  PV— 
  

  

  tensity 
  of 
  the 
  motion 
  is 
  so 
  great 
  that 
  Ir 
  becomes 
  of 
  the 
  same 
  

   or 
  of 
  smaller 
  order 
  than 
  p, 
  then 
  we 
  may 
  in 
  (4) 
  divide 
  the 
  

   numerator 
  and 
  the 
  denominator 
  by 
  h 
  2 
  ; 
  so 
  « 
  has 
  risen 
  two 
  

  

  pq... 
  

  

  orders 
  of 
  oreatness. 
  This 
  is 
  exact 
  for 
  all 
  vibrations 
  of 
  

  

  relation 
  of 
  both 
  kinds 
  and 
  every 
  degree. 
  So 
  we 
  have 
  now 
  

  

  but 
  to 
  examine 
  the 
  function 
  P. 
  

  

  pq 
  v 
  

   The 
  function 
  P 
  may 
  rise 
  in 
  greatness 
  in 
  consequence 
  of 
  

  

  those 
  terms 
  appearing 
  in 
  P, 
  which 
  have 
  themselves 
  risen 
  in 
  

  

  pq... 
  

  

  greatness, 
  i. 
  e. 
  have 
  become 
  of 
  a 
  smaller 
  order 
  than 
  in 
  the 
  

   general 
  case 
  where 
  no 
  relation 
  exists. 
  This 
  will 
  not 
  occur 
  

   for 
  the 
  vibration 
  of 
  relation 
  of 
  the 
  first 
  kind 
  and 
  first 
  degree. 
  

   It 
  will 
  occur, 
  however, 
  for 
  the 
  vibration 
  of 
  the 
  second 
  kind 
  

   and 
  first 
  degree, 
  in 
  consequence 
  of 
  terms 
  as 
  e. 
  g. 
  

  

  cos 
  2 
  <f) 
  cos 
  ((i> 
  — 
  l)(\) 
  + 
  <jy\r+ 
  ). 
  

  

  This 
  term 
  having 
  risen 
  two 
  orders, 
  P 
  will 
  also 
  rise 
  two 
  

  

  p 
  + 
  lqr... 
  

  

  orders. 
  So 
  the 
  term 
  of 
  the 
  second 
  kind 
  and 
  first 
  degree 
  

   will 
  rise 
  four 
  orders, 
  while 
  that 
  of 
  the 
  first 
  kind 
  and 
  first 
  

   degree 
  will 
  rise 
  only 
  two 
  orders. 
  The 
  order 
  of 
  the 
  former 
  

   being 
  in 
  general 
  Si 
  + 
  1, 
  that 
  of 
  the 
  second 
  Sx 
  — 
  1, 
  both 
  terms 
  

   will 
  rise 
  to 
  the 
  same 
  order, 
  namely 
  S,— 
  3. 
  

  

  Now 
  we 
  pass 
  on 
  to 
  the 
  vibrations 
  of 
  relation 
  of 
  the 
  second 
  

   _ 
  e. 
  In 
  the 
  P 
  function 
  for 
  the 
  term 
  of 
  the 
  first 
  kind 
  

  

  