﻿al'out 
  a 
  Position 
  of 
  Equilibrium, 
  307 
  

  

  envelope. 
  From 
  the 
  latter 
  one 
  it 
  follows 
  that 
  either 
  cos 
  2^ 
  

   or 
  cos 
  2t 
  2 
  is 
  positive. 
  Let 
  cos 
  2t 
  x 
  be 
  positive, 
  then 
  cos 
  2t 
  x 
  

   lies 
  between 
  £ 
  and 
  1, 
  cos 
  2t 
  2 
  between 
  — 
  -| 
  and 
  — 
  1. 
  

  

  A 
  Lissajous 
  curve 
  corresponding 
  to 
  cos</> 
  = 
  consists 
  of 
  

   three 
  parts, 
  which 
  are 
  separated 
  by 
  the 
  two 
  double 
  points. 
  

   For 
  the 
  middle 
  part 
  x 
  lies 
  between 
  J^/f 
  anc 
  ^ 
  ~i\/% 
  » 
  

   therefore 
  cos 
  t 
  lies 
  for 
  this 
  part 
  between 
  J 
  and 
  —J, 
  and 
  

   cos 
  2t 
  between 
  —1 
  and 
  — 
  ^. 
  For 
  the 
  other 
  parts 
  cos 
  2t 
  lies 
  

   between 
  and 
  —1 
  or 
  — 
  J 
  and 
  +1. 
  

  

  Therefore 
  the 
  envelope 
  consists 
  of 
  a 
  part 
  that 
  envelopes 
  

   the 
  middle 
  part 
  of 
  the 
  Lissajous 
  figures 
  and 
  a 
  part 
  that 
  

   envelopes 
  the 
  other 
  parts. 
  

  

  § 
  40. 
  For 
  an 
  arbitrary 
  value 
  of 
  r 
  the 
  maximal 
  value 
  of 
  $ 
  

   belongs 
  to 
  f=J, 
  because 
  cf> 
  is 
  max., 
  when 
  cos 
  6 
  is 
  min., 
  but 
  

   from 
  f\/?U~*?) 
  cos 
  <j> 
  = 
  r 
  it 
  follows 
  that 
  cos 
  (/> 
  is 
  min., 
  when 
  

   £ 
  3 
  (l 
  — 
  f) 
  is 
  max., 
  and 
  this 
  takes 
  place 
  for 
  f=J. 
  

  

  The 
  locus 
  of 
  the 
  double 
  points 
  of 
  the 
  system 
  of 
  curves 
  may 
  

   be 
  determined 
  in 
  this 
  way. 
  For 
  a 
  definite 
  Lissajous 
  curve 
  

   the 
  coordinates 
  of 
  the 
  double 
  points 
  are 
  : 
  

  

  *= 
  ± 
  k/ft 
  y= 
  +W 
  1 
  - 
  £ 
  cos 
  9- 
  

  

  By 
  eliminating 
  J 
  and 
  <f> 
  between 
  these 
  two 
  equations 
  and 
  

   the 
  relation 
  %y/\(l 
  — 
  £) 
  cos 
  </> 
  = 
  ■?• 
  we 
  get: 
  

  

  On 
  this 
  curve 
  lie 
  also 
  the 
  summits 
  of 
  the 
  double 
  curves. 
  

  

  When 
  for 
  an 
  arbitrary 
  value 
  of 
  r 
  we 
  wish 
  to 
  determine 
  the 
  

   shape 
  of 
  the 
  envelope, 
  we 
  have 
  to 
  consider 
  that 
  this 
  envelope 
  

   has 
  no 
  points 
  where 
  the 
  tangential 
  line 
  is 
  parallel 
  to 
  the 
  

   Y-axis, 
  and 
  that 
  the 
  only 
  points 
  where 
  the 
  tangential 
  line 
  

   is 
  parallel 
  to 
  the 
  X-axis 
  are 
  the 
  summits 
  of 
  the 
  double 
  

   carves 
  (conf. 
  § 
  32; 
  here 
  MQ=y/l 
  / 
  2 
  (£) 
  = 
  VW; 
  

   f 
  does 
  not 
  become 
  or 
  1). 
  

  

  Besides 
  the 
  ends 
  of 
  the 
  double 
  curves 
  (with 
  the 
  direction 
  

   of 
  the 
  tangential 
  lines) 
  and 
  the 
  summits 
  of 
  the 
  double 
  

   curves, 
  some 
  other 
  points 
  of 
  the 
  envelope 
  may 
  easily 
  be 
  

   found. 
  We 
  therefore 
  must 
  consider 
  the 
  Lissajous 
  curve, 
  

   for 
  which 
  f=#. 
  

  

  (22; 
  of 
  p. 
  298 
  takes 
  this 
  form 
  : 
  

  

  3(l-{) 
  sin 
  (?jl-(f>) 
  sin 
  (*-0)-fsin 
  3t 
  sin 
  < 
  = 
  0. 
  (30) 
  

   For 
  f=f 
  we 
  get 
  

  

  sin 
  (:)t 
  — 
  tf>) 
  sin 
  (( 
  — 
  <(>)— 
  &in 
  ot 
  sin 
  t 
  = 
  0. 
  

  

  