﻿320 
  Mr, 
  H. 
  J. 
  E. 
  Beth 
  on 
  the 
  Oscillations 
  

  

  q 
  = 
  l. 
  r 
  = 
  0. 
  Periodic 
  form 
  of 
  motion 
  in 
  

  

  an 
  arbitrary 
  straight 
  line 
  

   passing 
  through 
  (fig. 
  41). 
  

   — 
  iq<r<0. 
  One 
  system 
  of 
  ellipses 
  ; 
  the 
  

   envelope 
  consists 
  of 
  two 
  

   circles 
  (fig. 
  42). 
  

   y= 
  — 
  \q. 
  Periodic 
  form 
  of 
  motion 
  in 
  a 
  

   circle; 
  £=-£, 
  cos 
  </> 
  = 
  (fig. 
  

   43). 
  

  

  l<q<x>. 
  r 
  = 
  0. 
  Periodic 
  form 
  of 
  motion; 
  

  

  J=0 
  or 
  f=l 
  (fig. 
  44). 
  

   —-^(q 
  — 
  T)<r<0. 
  Two 
  systems 
  of 
  ellipses 
  (fig. 
  

   45). 
  

   r= 
  —^(q 
  — 
  1). 
  Asymptotic 
  form 
  of 
  motion 
  

   (fig. 
  46). 
  

   — 
  \q<r< 
  — 
  -J(9 
  — 
  1) 
  . 
  One 
  system 
  of 
  ellipses 
  (figs. 
  

   47, 
  48, 
  49). 
  

   r= 
  —\q. 
  Periodic 
  form 
  of 
  motion 
  in 
  a 
  

   circle 
  ; 
  f=£, 
  cos 
  <£ 
  = 
  (fig. 
  

   50). 
  

  

  <2 
  = 
  oo. 
  r 
  = 
  0. 
  Periodic 
  form 
  of 
  motion; 
  

  

  f=0 
  or 
  f=l 
  (fig. 
  51). 
  

   — 
  \q<r<0. 
  Two 
  systems 
  of: 
  ellipses; 
  

   each 
  of 
  them 
  has 
  a 
  rectangle 
  

   as 
  envelope 
  (fig. 
  52). 
  

   r=.—\q* 
  The 
  envelope 
  of 
  the 
  system 
  

   of 
  ellipses 
  is 
  a 
  square 
  (fig. 
  

   53). 
  

  

  Mechanism 
  with 
  three 
  degrees 
  of 
  freedom. 
  

  

  § 
  54. 
  The 
  coordinates 
  of 
  the 
  representative 
  point 
  are 
  

   given 
  by 
  : 
  

  

  y 
  =v^cos(ny* 
  + 
  2ny# 
  

  

  y;» 
  

  

  % 
  

  

  ^=^cos(V 
  + 
  2n 
  2 
  /3J. 
  

  

  These 
  are 
  the 
  equations 
  of 
  the 
  osculating 
  curves. 
  By 
  

   eliminating 
  t 
  between 
  these 
  equations 
  two 
  by 
  two 
  we 
  find 
  

   the 
  projections 
  o£ 
  the 
  osculating 
  curves 
  on 
  the 
  planes 
  of 
  

   coordinates. 
  These 
  projections 
  are 
  Lissajous 
  curves 
  ; 
  the 
  

  

  