﻿about 
  a 
  Position 
  of 
  Equilibrium, 
  28' 
  

  

  found. 
  The 
  problem 
  is 
  reduced 
  to 
  quadratures. 
  It 
  is 
  easy 
  

   to 
  see 
  that 
  in 
  the 
  cases 
  Si=3 
  and 
  Si=4 
  the 
  coordinates 
  

   with 
  the 
  help 
  of 
  elliptic 
  functions 
  can 
  be 
  expressed 
  in 
  terms 
  

   of 
  the 
  time 
  *. 
  

  

  § 
  19. 
  In 
  § 
  5 
  (p. 
  275) 
  we 
  have 
  found 
  that 
  in 
  the 
  case 
  

   Si=3 
  the 
  coordinates 
  which 
  appear 
  in 
  the 
  relation 
  behave 
  

   as 
  if 
  they 
  were 
  the 
  only 
  coordinates 
  ; 
  the 
  remaining 
  co- 
  

   ordinates 
  behave 
  as 
  if 
  no 
  relation 
  existed. 
  

  

  This 
  is 
  not 
  true 
  for 
  the 
  cases 
  Si 
  = 
  4 
  and 
  ^ 
  x 
  — 
  2. 
  In 
  these 
  

   cases 
  the 
  function 
  R 
  contains 
  in 
  general 
  all 
  ct's. 
  So 
  all 
  /9V 
  

   are 
  functions 
  of 
  t. 
  The 
  function 
  II 
  contains 
  only 
  the 
  /3\s 
  

   related 
  to 
  those 
  coordinates 
  which 
  enter 
  in 
  the 
  relation. 
  

   Therefore 
  the 
  a* 
  related 
  to 
  the 
  remaining 
  coordinates 
  are 
  

   constants. 
  These 
  coordinates 
  feel 
  the 
  influence 
  of 
  the 
  

   relation 
  only 
  in 
  their 
  phase. 
  The 
  coordinates 
  which 
  appear 
  

   in 
  the 
  relation 
  feel 
  also 
  the 
  influence 
  of 
  the 
  remaining 
  

   coordinates 
  ; 
  this 
  influence 
  consists 
  in 
  a 
  modification 
  of 
  the 
  

   frequencies. 
  

  

  § 
  20. 
  The 
  expressions 
  (15) 
  and 
  (16) 
  for 
  -, 
  show 
  us 
  that 
  

  

  -j~ 
  t 
  /and, 
  according 
  to 
  (11), 
  also 
  — 
  , 
  ~, 
  J 
  are 
  zero 
  

  

  when 
  sin 
  <£ 
  = 
  (). 
  So 
  in 
  general 
  the 
  us 
  will 
  vary 
  periodically 
  

   between 
  two 
  limits. 
  The 
  extreme 
  values 
  of 
  the 
  as 
  will 
  be 
  

   known 
  when 
  the 
  values 
  of 
  fare 
  known 
  for 
  which 
  sin 
  </> 
  = 
  (). 
  

   The 
  equation, 
  from 
  which 
  these 
  values 
  of 
  f 
  are 
  found, 
  is 
  ob- 
  

   tained 
  by 
  putting 
  in 
  (14) 
  cos 
  <f> 
  equal 
  to 
  + 
  1. 
  This 
  equation 
  is 
  

   in 
  the 
  case 
  Sx 
  = 
  3 
  of 
  the 
  third, 
  in 
  the 
  cases 
  Si 
  = 
  4 
  and 
  S 
  x 
  =2 
  of 
  

  

  the 
  fourth 
  degree. 
  In 
  the 
  case 
  S 
  x 
  =2, 
  -£ 
  may 
  also 
  become 
  

  

  zero 
  tor 
  cos 
  o=— 
  '".<;-. 
  

   2/i(£) 
  

  

  5 
  21. 
  We 
  may 
  remark 
  now 
  that 
  the 
  motion 
  of 
  the 
  mech- 
  

   anism, 
  in 
  case 
  an 
  approximate 
  relation 
  between 
  the 
  principal 
  

   frequencies 
  exists, 
  is 
  wholly 
  different 
  from 
  the 
  general 
  case 
  

   in 
  which 
  no 
  relation 
  exists. 
  In 
  the 
  latter 
  case 
  at 
  first 
  

   approximation 
  the 
  principal 
  coordinates 
  are 
  independent 
  of 
  

   one 
  another; 
  each 
  principal 
  coordinate 
  performs 
  an 
  harmonic 
  

   vibration 
  of 
  a 
  constant 
  amplitude 
  ; 
  in 
  order 
  to 
  take 
  into 
  

   account 
  the 
  terms 
  of 
  higher 
  order 
  we 
  have 
  but 
  to 
  add 
  to 
  the 
  

   motion 
  of 
  each 
  principal 
  coordinate 
  motions 
  of 
  a 
  very 
  small 
  

   amplitude 
  (if 
  compared 
  with 
  the 
  principal 
  vibration 
  itself), 
  ^n 
  

  

  * 
  In 
  my 
  dissertation 
  (Amsterdam, 
  1910) 
  these 
  calculations 
  have 
  been 
  

   e^ec'ited 
  for 
  the 
  relation 
  2n 
  x 
  — 
  «y=p. 
  

  

  