﻿about 
  a 
  Position 
  of 
  Equilibrium. 
  273 
  

  

  Si=2. 
  We 
  shall 
  examine 
  how 
  by 
  the 
  decrease 
  of 
  the 
  

   amplitudes, 
  or. 
  what 
  is 
  the 
  same, 
  by 
  the 
  increase 
  of 
  p, 
  the 
  

   transition 
  to 
  the 
  general 
  case 
  takes 
  place. 
  

  

  The 
  difference 
  between 
  the 
  cases 
  Si>4 
  and 
  Sx<4 
  is 
  one 
  

  

  of 
  the 
  results 
  Prof. 
  Korteweg 
  arrives 
  at. 
  In 
  the 
  foregoing- 
  

   lines 
  I 
  have 
  tried 
  to 
  repeat 
  the 
  main 
  points 
  of 
  his 
  reasoning; 
  

   however, 
  I 
  have 
  omitted 
  what 
  seemed 
  to 
  me 
  to 
  be 
  of 
  less 
  

   importance 
  for 
  what 
  will 
  follow 
  now. 
  So 
  Prof. 
  Korteweg 
  

   finds 
  that 
  there 
  exists 
  a 
  third 
  kind 
  of 
  vibrations 
  of 
  relation, 
  

   whose 
  frequencies 
  lie 
  generally 
  in 
  the 
  neighbourhood 
  of 
  all 
  

   principal 
  frequencies, 
  which 
  do 
  not 
  appear 
  in 
  the 
  relation 
  ; 
  

   they 
  behave 
  as 
  the 
  vibrations 
  of 
  the 
  second 
  kind. 
  

  

  The 
  investigation 
  of 
  the 
  motion 
  of 
  a 
  vibrating 
  mechanism 
  

   having 
  two 
  degrees 
  of 
  freedom 
  in 
  the 
  cases 
  where 
  Prof. 
  

   Korteweg's 
  developments 
  cease 
  to 
  represent 
  this 
  motion 
  was 
  

   the 
  subject 
  of 
  my 
  dissertation 
  (Amsterdam, 
  1910). 
  How- 
  

   ever, 
  Prof. 
  Korteweg 
  had 
  found 
  already 
  the 
  nature 
  of 
  the 
  

   motion 
  for 
  the 
  case 
  of 
  the 
  strict 
  relation 
  n 
  =2n 
  x 
  and 
  the 
  

   relation 
  between 
  f 
  and 
  cf> 
  for 
  this 
  case, 
  which 
  we 
  shall 
  deduce 
  

   ]>. 
  286, 
  as 
  well 
  as 
  the 
  special 
  cases 
  which 
  for 
  this 
  relation 
  may 
  

   occur. 
  This 
  solution 
  has 
  been 
  my 
  key 
  to 
  the 
  examination 
  

   of 
  the 
  other 
  eases, 
  though 
  I 
  have 
  during 
  the 
  course 
  of 
  my 
  in- 
  

   vestigations 
  directed 
  them 
  along 
  other 
  lines 
  than 
  were 
  traced 
  

   out 
  in 
  Prof. 
  Korteweg 
  s 
  solution. 
  The 
  dissertation 
  mentioned 
  

   before 
  was 
  published 
  in 
  an 
  abridged 
  form 
  (Proceedings, 
  

   Amsterdam, 
  pp. 
  618-635, 
  and 
  pp. 
  735-750, 
  1910; 
  Archives 
  

   Neerlandaises, 
  series 
  2, 
  vol. 
  xv. 
  pp. 
  246-283, 
  1910). 
  In 
  

   another 
  paper 
  (Proceedings, 
  Amsterdam, 
  pp. 
  742-761, 
  1911 
  ; 
  

   Archives 
  N6erlandaises, 
  series 
  3 
  a, 
  vol. 
  i. 
  pp. 
  185-208, 
  1912) 
  

   I 
  have 
  extended 
  the 
  investigation 
  to 
  a 
  mechanism 
  having 
  an 
  

   arbitrary 
  number 
  of 
  degrees 
  of 
  freedom. 
  Some 
  months 
  ago 
  

   in 
  a 
  prize-essay 
  (not 
  yet 
  published) 
  written 
  in 
  answer 
  to 
  a 
  

   problem 
  set 
  by 
  the 
  Mathematical 
  Society 
  of 
  Amsterdam, 
  I 
  

   have 
  examined 
  the 
  envelope 
  of 
  systems 
  of 
  Lissajous' 
  curves 
  

   which 
  occur 
  in 
  the 
  question 
  we 
  have 
  in 
  view 
  ; 
  in 
  my 
  disser- 
  

   tation 
  the 
  envelope 
  had 
  been 
  found 
  only 
  for 
  the 
  case 
  

   n 
  ='2 
  1' 
  +p. 
  The 
  subject 
  having 
  come 
  so 
  far 
  to 
  a 
  conclusion 
  

   it 
  may 
  appear 
  desirable 
  to 
  publish 
  a 
  new 
  summary 
  of 
  the 
  

   problem. 
  

  

  § 
  4. 
  In 
  cas^ 
  a 
  relation 
  of 
  the 
  form 
  

  

  y;". 
  + 
  yi" 
  + 
  = 
  p 
  

  

  exists, 
  some 
  terms 
  in 
  the 
  equations 
  of 
  motion, 
  after 
  sub- 
  

   stitution 
  of 
  the 
  expressions 
  which 
  represent 
  in 
  the 
  general 
  case 
  

  

  the 
  coordinate- 
  at 
  first 
  approximation, 
  give 
  rise 
  to 
  terms 
  of 
  

  

  