﻿306 
  Mr. 
  H. 
  J. 
  E. 
  Beth 
  on 
  the 
  Oscillations 
  

  

  For 
  an 
  arbitrary 
  value 
  of 
  r 
  we 
  have 
  the 
  case 
  of 
  libration, 
  

   cos 
  </> 
  remaining 
  positive. 
  The 
  maximal 
  value 
  {^q\/3) 
  and 
  

   the 
  minimal 
  value 
  of 
  r 
  (0) 
  give 
  rise 
  to 
  the 
  special 
  cases 
  

   resp. 
  of 
  the 
  periodic 
  and 
  of 
  the 
  asymptotic 
  form 
  of 
  motion. 
  

  

  In 
  the 
  case 
  of 
  the 
  periodic 
  form 
  of 
  motion 
  ?=f 
  and 
  

   sin<£ 
  = 
  0; 
  the 
  same 
  double 
  curve 
  is 
  continually 
  described, 
  

   which 
  is 
  in 
  the 
  same 
  time 
  the 
  envelope 
  (fig. 
  20) 
  . 
  

  

  In 
  the 
  case 
  of 
  the 
  asymptotic 
  form 
  of 
  motion 
  we 
  have 
  

  

  77* 
  

  

  <£= 
  ~ 
  . 
  For 
  this 
  case 
  (22) 
  of 
  p. 
  298 
  passes 
  into 
  : 
  

  

  3(1 
  — 
  f) 
  cos 
  St 
  cos 
  t—% 
  sin 
  St 
  sin 
  £ 
  = 
  0. 
  . 
  . 
  (29) 
  

   Therefore 
  

  

  ,,_ 
  3 
  cos 
  St 
  cos 
  t 
  

  

  3 
  cos 
  St 
  cos 
  t 
  + 
  sin 
  St 
  sin 
  t' 
  

  

  By 
  substitution 
  of 
  this 
  in 
  the 
  expressions 
  for 
  x 
  and 
  y 
  9 
  

   we 
  find 
  

  

  _ 
  / 
  3 
  cos 
  St 
  cos 
  3 
  1 
  ^ 
  _1 
  / 
  

  

  V 
  3 
  cos 
  St 
  cos 
  t 
  + 
  sin 
  St 
  sin 
  t 
  ' 
  ~~ 
  3 
  v 
  i 
  

  

  sin 
  t 
  sin 
  3 
  St 
  

  

  3 
  cos 
  St 
  cos 
  t 
  + 
  sin 
  St 
  sin 
  t' 
  

   Furthermore 
  we 
  have 
  : 
  

  

  dy 
  sj\ 
  — 
  £cos3£ 
  / 
  sin 
  St 
  sin 
  t 
  cos 
  2 
  St 
  _ 
  / 
  sin 
  6t 
  

  

  da;"" 
  \Z'X 
  sin 
  t 
  V 
  3 
  cos 
  St 
  cos 
  t 
  sin 
  2 
  1 
  ~ 
  v 
  3 
  sin 
  2t" 
  

  

  From 
  this 
  it 
  is 
  possible 
  to 
  deduce 
  the 
  shape 
  of 
  the 
  envelope. 
  

   x 
  = 
  for 
  cos 
  3£ 
  = 
  and 
  for 
  cos 
  £ 
  = 
  0. 
  For 
  cos 
  3^ 
  = 
  0, 
  

  

  y=±&ji= 
  '> 
  for 
  cos 
  £ 
  = 
  0, 
  y=±i^=±l. 
  y 
  = 
  for 
  

  

  sin 
  St 
  =0 
  and 
  for 
  sin 
  t 
  = 
  0. 
  For 
  sin 
  St 
  = 
  0, 
  x= 
  ± 
  J, 
  -f- 
  =0; 
  

  

  for 
  sin£ 
  = 
  0, 
  x= 
  + 
  l,-f- 
  =±1- 
  The 
  enveloping 
  curve 
  is, 
  

  

  besides 
  some 
  osculating 
  curves, 
  represented 
  in 
  fig. 
  22. 
  

  

  Now 
  we 
  may 
  put 
  the 
  question, 
  what 
  is 
  the 
  signification 
  

   of 
  the 
  different 
  parts 
  of 
  the 
  envelope. 
  A 
  definite 
  Lissajous 
  

   curve 
  is 
  enveloped 
  in 
  different 
  points, 
  in 
  other 
  words 
  to 
  a 
  

   definite 
  value 
  of 
  f 
  correspond 
  different 
  values 
  of 
  t 
  for 
  the 
  

   envelope. 
  Let 
  t 
  x 
  and 
  t 
  2 
  be 
  two 
  of 
  these 
  values, 
  then 
  from 
  

   (29) 
  it 
  follows 
  

  

  sin 
  3^ 
  sin 
  t 
  x 
  __ 
  sin 
  St 
  2 
  sin 
  t 
  2 
  

   cos 
  3^ 
  cos 
  t 
  x 
  cos 
  St 
  2 
  cos 
  t 
  2 
  ' 
  

  

  After 
  reduction 
  this 
  relation 
  degenerates 
  into 
  

  

  cos 
  2^ 
  = 
  cos 
  2t 
  2 
  and 
  cos 
  2t 
  l 
  cos 
  2t 
  

  

  ,— 
  h 
  

  

  The 
  former 
  equation 
  expresses 
  only 
  that 
  is 
  centre 
  of 
  the 
  

  

  