﻿288 
  Mr. 
  H. 
  J. 
  E. 
  Beth 
  on 
  the 
  Oscillations 
  

  

  the 
  former 
  case 
  the 
  principal 
  coordinates 
  are 
  no 
  longer 
  in- 
  

   dependent 
  of 
  each 
  other 
  ; 
  the 
  amplitudes 
  of 
  the 
  principal 
  

   vibrations 
  vary 
  with 
  the 
  time 
  ; 
  so 
  the 
  quantities 
  of 
  energy 
  

   belonging 
  to 
  each 
  of 
  the 
  coordinates 
  vary 
  also 
  with 
  the 
  time 
  ; 
  

   (12, 
  p. 
  285) 
  expresses 
  that 
  the 
  whole 
  of 
  the 
  energy 
  is 
  

   constant. 
  

  

  When 
  a 
  relation 
  exists, 
  then 
  the 
  form 
  of 
  motion 
  of 
  the 
  

   mechanism 
  will 
  constantly 
  alter, 
  because 
  the 
  a's 
  and 
  the 
  /3's 
  

   vary 
  with 
  the 
  time. 
  However, 
  after 
  a 
  certain 
  lapse 
  of 
  time 
  

   the 
  configuration 
  will 
  be 
  the 
  same 
  as 
  before. 
  In 
  order 
  to 
  

   investigate 
  the 
  forms 
  of 
  motion, 
  which 
  will 
  successively 
  

   occur, 
  it 
  will 
  be 
  sufficient 
  to 
  discuss 
  the 
  relation 
  between 
  f 
  

   and 
  (j). 
  To 
  this 
  aim 
  we 
  shall 
  represent 
  this 
  relation 
  in 
  polar 
  

   coordinates 
  with 
  (j> 
  as 
  polar 
  angle, 
  f 
  (or 
  v'f 
  or 
  VI 
  — 
  f) 
  as 
  

   radius 
  vector. 
  The 
  curves 
  are 
  symmetrical 
  with 
  respect 
  to 
  

   the 
  origin 
  of 
  angles. 
  The 
  form 
  of 
  the 
  curves 
  changes 
  with 
  

   the 
  value 
  of 
  the 
  coefficients 
  in 
  the 
  second 
  member 
  of 
  the 
  

   equation. 
  The 
  intersecting 
  points 
  with 
  the 
  origin 
  of 
  angles 
  

   are 
  given 
  by 
  sin$ 
  = 
  0. 
  Now 
  there 
  are, 
  letting 
  alone 
  the 
  

   special 
  cases, 
  two 
  different 
  cases 
  : 
  

  

  1°. 
  For 
  the 
  extreme 
  values 
  of 
  £, 
  c/> 
  has 
  the 
  same 
  value 
  ; 
  

   then 
  (j) 
  varies 
  between 
  two 
  limits. 
  Case 
  of 
  libration. 
  

  

  2°. 
  For 
  the 
  extreme 
  values 
  of 
  f, 
  (/> 
  is 
  one 
  time 
  0, 
  one 
  time 
  

   7T 
  ; 
  cf) 
  takes 
  all 
  values. 
  General 
  case. 
  

  

  It 
  is 
  clear 
  that 
  there 
  are 
  two 
  special 
  cases. 
  The 
  one 
  is 
  to 
  

   be 
  considered 
  as 
  the 
  limit 
  case 
  of 
  the 
  libration 
  ; 
  the 
  extreme 
  

   values 
  of 
  f 
  have 
  coincided 
  ; 
  sin 
  <fi 
  (or 
  cos 
  </>) 
  is 
  zero 
  invariably. 
  

   The 
  form 
  of 
  motion 
  does 
  not 
  alter. 
  Periodic 
  case. 
  

  

  To 
  a 
  same 
  value 
  of 
  the 
  coefficients 
  two 
  closed 
  curves 
  may 
  

   correspond. 
  For 
  a 
  special 
  value 
  of 
  the 
  coefficients 
  these 
  curves 
  

   may 
  have 
  a 
  common 
  point. 
  Then 
  the 
  motion 
  approaches 
  

   asymptotically 
  to 
  the 
  form 
  of 
  motion 
  indicated 
  by 
  this 
  point. 
  

   Asymptotic 
  case. 
  

  

  § 
  22. 
  We 
  shall 
  apply 
  this 
  to 
  the 
  different 
  cases. 
  

  

  Si 
  = 
  3. 
  

  

  2n 
  x 
  — 
  n 
  t/ 
  = 
  p. 
  We 
  suppose 
  first 
  p 
  = 
  ; 
  then 
  we 
  have 
  the 
  

   ease 
  of 
  the 
  strict 
  relation. 
  The 
  relation 
  between 
  f 
  and 
  <£ 
  

   ^uns 
  : 
  

  

  f 
  Vl 
  — 
  f 
  cos 
  (j) 
  = 
  k 
  ( 
  V 
  1 
  — 
  f 
  we 
  take 
  as 
  radius 
  vector). 
  

  

  For 
  k>0 
  the 
  f-<£ 
  curves 
  lie 
  to 
  the 
  right 
  of 
  the 
  straight 
  

  

  line 
  </)= 
  — 
  for 
  k<0 
  to 
  the 
  left 
  of 
  it 
  ; 
  k 
  = 
  furnishes 
  de- 
  

  

  generation 
  into 
  the 
  straight 
  line 
  (j>= 
  -^> 
  the 
  point 
  f=l 
  and 
  

  

  