﻿308 
  Mr. 
  II. 
  J. 
  E. 
  Beth 
  on 
  the 
  Oscillations 
  

  

  After 
  reduction 
  : 
  

  

  The 
  eight 
  points 
  in 
  which 
  the 
  curve 
  we 
  have 
  in 
  view 
  is 
  

   enveloped 
  are 
  

  

  x 
  = 
  J^/d 
  cos 
  ( 
  J0 
  + 
  nx 
  45°), 
  

  

  ?/ 
  = 
  icos(~i^ 
  + 
  3nx 
  45°)= 
  ±^cos 
  (1^ 
  + 
  ^x45°). 
  

  

  Therefore 
  - 
  = 
  ±3^3. 
  

  

  y 
  

  

  These 
  points 
  are 
  situated 
  on 
  the 
  diagonals 
  of 
  the 
  rectangle 
  

   in 
  which 
  the 
  curve 
  is 
  described. 
  

   Furthermore, 
  

  

  dy 
  _ 
  x 
  /l-£sin(3*--<fr) 
  _ 
  + 
  1 
  „ 
  

   dx 
  ^/fsin*. 
  -3V 
  • 
  

  

  The 
  tangential 
  lines 
  in 
  these 
  points 
  form 
  with 
  the 
  X-axis 
  

   angles 
  of 
  30° 
  or 
  150°. 
  

  

  In 
  the 
  envelope 
  cusps 
  appear. 
  In 
  order 
  to 
  determine 
  the 
  

   situation 
  of 
  these 
  cusps, 
  we 
  would 
  have 
  to 
  calculate 
  f, 
  <£> 
  

   and 
  t 
  from 
  three 
  equations, 
  namely 
  (28) 
  of 
  § 
  39 
  ; 
  (30) 
  of 
  

  

  this 
  section, 
  and 
  the 
  third 
  equation 
  is 
  (§ 
  33) 
  either 
  j 
  = 
  co 
  

  

  dx 
  

   or 
  — 
  = 
  0. 
  However, 
  these 
  systems 
  become 
  rather 
  intricate. 
  

  

  In 
  fig. 
  21 
  the 
  envelope 
  with 
  some 
  of 
  the 
  Lissajous 
  curves 
  

   is 
  represented. 
  

  

  § 
  41. 
  The 
  Lissajous 
  curves 
  for 
  this 
  case, 
  given 
  by 
  

  

  g==VfcosE, 
  Y 
  

  

  y=VWcos(*-0)J 
  " 
  ' 
  ' 
  l 
  / 
  

  

  are 
  ellipses. 
  The 
  double 
  curves 
  are 
  straight 
  lines 
  passing 
  

   through 
  0. 
  The 
  ellipses 
  corresponding 
  to 
  cosc/> 
  = 
  have 
  

   their 
  axes 
  along 
  the 
  axes 
  of 
  coordinates. 
  For 
  an 
  arbitrary 
  

   value 
  of 
  cf) 
  the 
  ellipse 
  has 
  an 
  arbitrary 
  shape 
  and 
  position. 
  

  

  In 
  order 
  to 
  investigate 
  the 
  change 
  in 
  shape 
  and 
  position 
  

   of 
  the 
  ellipse, 
  we 
  may 
  write 
  down 
  the 
  well-known 
  relations 
  

   which 
  may 
  serve 
  for 
  the 
  calculation 
  of 
  the 
  axes 
  of 
  the 
  

   ellipse 
  and 
  the 
  angle 
  of 
  inclination 
  of 
  the 
  long 
  axis 
  with 
  the 
  

   X-axis. 
  If 
  A 
  and 
  B 
  are 
  half 
  the 
  larger 
  and 
  half 
  the 
  smaller 
  

  

  