﻿300 
  Mr. 
  H. 
  J. 
  E. 
  Beth 
  on 
  the 
  Oscillations 
  

  

  In 
  order 
  to 
  examine 
  the 
  conditions 
  ~ 
  - 
  = 
  co 
  and 
  -j- 
  = 
  oo 
  

  

  dt 
  at 
  

  

  we 
  write 
  (20) 
  in 
  the 
  form 
  

  

  /(<!>, 
  Z,t)=o. 
  

  

  From 
  this 
  equation 
  and 
  (19) 
  it 
  follows 
  : 
  

  

  d</>'dt 
  ^ 
  ^z'dt 
  + 
  b* 
  ~ 
  u 
  * 
  

  

  In 
  general 
  the 
  two 
  conditions 
  -~ 
  -co 
  and 
  -^ 
  =ao 
  

   coincide 
  ; 
  they 
  are 
  satisfied 
  when 
  

  

  |.sin^-|. 
  t 
  '(r)=0. 
  

  

  2?7 
  T 
  — 
  W 
  y 
  = 
  p. 
  

  

  § 
  34. 
  The 
  osculating 
  curves 
  are 
  given 
  by 
  

  

  #= 
  V£cos£, 
  

   y=i\/l-|'cos(2^-^), 
  

  

  ar= 
  Vfcosf, 
  \ 
  

  

  whilst 
  

  

  ? 
  i/I^fcos 
  <£ 
  = 
  & 
  + 
  />'£. 
  (24) 
  

  

  The 
  vertices 
  of 
  the 
  circumscribed 
  rectangles 
  lie 
  in 
  the 
  cir- 
  

   cumference 
  of 
  the 
  ellipse 
  # 
  2 
  + 
  4?/ 
  2 
  = 
  l. 
  

  

  The 
  Lissajous 
  double 
  curves 
  are 
  parabolae 
  ; 
  for 
  </> 
  = 
  the 
  

   parabola 
  has 
  its 
  opening 
  turned 
  upwards, 
  for 
  <£ 
  = 
  it 
  turned 
  

   downwards. 
  

  

  The 
  Lissajous 
  curve 
  corresponding 
  to 
  cos 
  </> 
  = 
  has 
  a 
  double 
  

   point 
  in 
  0. 
  

  

  The 
  curves 
  corresponding 
  to 
  an 
  arbitrary 
  value 
  of 
  </> 
  have 
  

   a 
  node 
  in 
  the 
  Y-axis 
  at 
  a 
  distance 
  — 
  ^ 
  Vl 
  — 
  f 
  cos</> 
  from 
  the 
  

   X-axis. 
  

  

  In 
  the 
  case 
  of 
  libration 
  the 
  curves 
  of 
  the 
  system 
  have 
  

   their 
  nodes 
  all 
  at 
  the 
  same 
  side 
  of 
  ; 
  the 
  double 
  curves 
  have 
  

   their 
  opening 
  to 
  the 
  same 
  side. 
  In 
  the 
  case 
  of 
  the 
  general 
  

   form 
  of 
  motion 
  the 
  nodes 
  of 
  the 
  curves 
  lie 
  partly 
  on 
  one, 
  

   partly 
  on 
  the 
  other 
  side 
  of 
  ; 
  the 
  double 
  curves 
  have 
  their 
  

   openings 
  to 
  different 
  sides. 
  In 
  the 
  case 
  of 
  the 
  periodic 
  form 
  

   of 
  motion 
  the 
  same 
  parabola 
  is 
  continually 
  described. 
  In 
  

   the 
  asymptotic 
  case 
  the 
  motion 
  approaches 
  to 
  a 
  Y-vibration. 
  

  

  