﻿26S 
  ] 
  

  

  XVIII. 
  The 
  Oscillations 
  about 
  a 
  Position 
  of 
  Equilibrium 
  where 
  

   a 
  Simple 
  Linear 
  Relation 
  exists 
  between 
  the 
  Frequencies 
  of 
  

   the 
  Principal 
  Vibrations. 
  By 
  H. 
  J. 
  E. 
  Beth 
  *. 
  

  

  [Plates 
  VII. 
  & 
  Vin.] 
  

  

  §1. 
  X?OR 
  a 
  mechanism 
  with 
  m 
  degrees 
  of 
  freedom 
  per- 
  

   JL 
  forming 
  small 
  oscillations 
  about 
  a 
  position 
  of 
  

   stable 
  equilibrium 
  the 
  equations 
  of 
  motion 
  may 
  be 
  written 
  

   in 
  the 
  form 
  : 
  

  

  '£ 
  + 
  n 
  x 
  2 
  x 
  4- 
  tic 
  2 
  + 
  %xy 
  + 
  %xx 
  + 
  txy 
  + 
  tx 
  2 
  + 
  Ixy 
  + 
  Xxx 
  2 
  + 
  

  

  9 
  9^9 
  9^9 
  9 
  9 
  

  

  y 
  -f- 
  ihfy 
  + 
  Xx 
  2 
  + 
  . 
  . 
  . 
  . 
  

  

  9 
  

  

  = 
  [• 
  (i) 
  

  

  where 
  x, 
  y, 
  etc. 
  are 
  the 
  principal 
  coordinates 
  of 
  the 
  mechan- 
  

   ism 
  ; 
  they 
  vanish 
  in 
  the 
  position 
  of 
  equilibrium 
  ; 
  %ic 
  2 
  is 
  

  

  •0-9 
  9 
  

  

  written 
  instead 
  of 
  ax 
  2 
  -\-by 
  2 
  -\- 
  cz 
  2 
  -\- 
  . 
  . 
  . 
  . 
  etc.; 
  n 
  x 
  , 
  n 
  y 
  , 
  etc. 
  are 
  

   the 
  frequencies 
  of 
  the 
  principal 
  vibrations. 
  

  

  In 
  his 
  treatise 
  t 
  : 
  " 
  On 
  certain 
  vibrations 
  of 
  higher 
  order 
  

   of 
  abnormal 
  intensity 
  (vibrations 
  of 
  relation) 
  for 
  mechanisms 
  

   with 
  more 
  degrees 
  of 
  freedom 
  " 
  ( 
  Verhandelingen 
  der 
  Konink- 
  

   lyke 
  Alcademie 
  van 
  Wetensckappen, 
  vol. 
  v. 
  No. 
  8, 
  1897 
  ; 
  

   Archives 
  Neerlandaises, 
  vol. 
  i. 
  series 
  2, 
  pp. 
  229-260), 
  Prof. 
  

   Korteweg 
  has 
  shown 
  that 
  the 
  coordinates 
  x, 
  y 
  .... 
  may 
  be 
  

   expressed 
  by 
  the 
  following 
  expansions 
  in 
  series 
  : 
  

  

  (2) 
  (3) 
  (3) 
  

  

  X 
  = 
  a. 
  + 
  Ah 
  COS 
  (j) 
  + 
  u 
  cos 
  ^ 
  + 
  a 
  COS 
  £ 
  + 
  ..... 
  

  

  00.. 
  .0 
  010. 
  ..0 
  C01...0 
  

  

  ^ 
  (S) 
  

  

  -f 
  2 
  « 
  cos 
  (j><f> 
  + 
  qyjr 
  + 
  rx 
  + 
  . 
  . 
  

  

  pqr... 
  

  

  (2) 
  (3) 
  (3) 
  

  

  y 
  = 
  #+ 
  ft 
  COS 
  (/> 
  + 
  B7i 
  COS 
  ^-f 
  /3 
  COS 
  % 
  + 
  . 
  . 
  . 
  . 
  

   00-. 
  .0 
  100.. 
  .0 
  001. 
  ..0 
  

  

  (S) 
  

   + 
  2 
  /3 
  cos 
  (j9<£ 
  + 
  2^r 
  + 
  r% 
  + 
  . 
  . 
  

  

  1 
  

  

  (2> 
  

  

  * 
  Communicated 
  by 
  the 
  Author. 
  

  

  t 
  " 
  Over 
  zekere 
  trillirigen 
  van 
  hooger 
  orde 
  van 
  abnormale 
  intensiteit. 
  

   (relatietrillingen) 
  bij 
  mechanismen 
  met 
  meerdere 
  graden 
  van 
  vrijheid." 
  

  

  