﻿296 
  Mr. 
  H. 
  J. 
  E. 
  Beth 
  on 
  the 
  Oscillations 
  

  

  the 
  curve 
  corresponding 
  to 
  a 
  value 
  f+ 
  Af 
  for 
  f. 
  So 
  we 
  have 
  

   to 
  eliminate 
  Af, 
  A</>, 
  and 
  A£ 
  between 
  : 
  

  

  The 
  result 
  is 
  the 
  following 
  equation 
  : 
  

  

  If 
  we 
  eliminate 
  f, 
  </>, 
  and 
  * 
  between 
  this 
  equation 
  and 
  the 
  

   equations 
  written 
  down 
  at 
  the 
  beginning 
  of 
  this 
  section, 
  the 
  

   result 
  is 
  the 
  equation 
  of 
  the 
  envelope. 
  

  

  § 
  30. 
  We 
  start 
  from 
  a 
  system 
  of 
  Lissajous 
  curves 
  which 
  

   is 
  more 
  general 
  than 
  the 
  system 
  (17) 
  of 
  § 
  27, 
  namely 
  

  

  *=/l(£)C0S*, 
  y=/ 
  2 
  (?)C0s(7*-<£), 
  COS0 
  = 
  l|r(f). 
  . 
  (19) 
  

  

  By 
  substitution 
  of 
  these 
  expressions 
  for 
  x 
  and 
  y, 
  the 
  equa- 
  

   tion 
  of 
  the 
  envelope 
  takes 
  this 
  form 
  : 
  

  

  y/sXD/i'(S) 
  C0S 
  t 
  sin(yf 
  — 
  0)sin 
  0— 
  /^/^(Qsin 
  £ 
  cos 
  (yt—</>) 
  sin 
  ^ 
  

  

  Vi®/2(?)sinisin( 
  7 
  ^-^)t'(D=0. 
  . 
  . 
  (20) 
  

  

  § 
  31. 
  This 
  equation 
  is 
  satisfied 
  by 
  sin 
  (f> 
  = 
  0, 
  sin 
  £ 
  = 
  0. 
  Now 
  

   the 
  double 
  curves 
  correspond 
  to 
  sin 
  c/> 
  = 
  ; 
  the 
  ends 
  of 
  these 
  

   curves 
  (meant 
  in 
  a 
  dynamical 
  sense) 
  correspond 
  to 
  sin£ 
  = 
  0. 
  

   Therefore 
  : 
  

  

  The 
  ends 
  of 
  the 
  double 
  curves 
  lie 
  on 
  the 
  envelope. 
  

  

  The 
  tangential 
  line 
  of 
  the 
  envelope 
  in 
  a 
  certain 
  point 
  

   coincides 
  with 
  the 
  tangential 
  line 
  of 
  the 
  osculating 
  curve, 
  

   which 
  in 
  the 
  point 
  considered 
  is 
  enveloped. 
  Therefore 
  we 
  

   have 
  for 
  the 
  envelope 
  : 
  

  

  dy 
  yfiiQsmiyt 
  — 
  (j>) 
  

  

  dx 
  /i(f)sin 
  t 
  

  

  (21) 
  

  

  dv 
  

   So 
  we 
  see 
  that 
  -7- 
  takes 
  the 
  indefinite 
  form 
  for 
  sin 
  £ 
  = 
  

   dx 
  

  

  (this 
  involves 
  sin 
  y£ 
  = 
  0) 
  and 
  sin 
  (j> 
  = 
  0, 
  in 
  other 
  words, 
  for 
  the 
  

  

  ends 
  of 
  the 
  double 
  curves. 
  We 
  have 
  now 
  to 
  investigate 
  in 
  

  

  what 
  manner 
  the 
  ends 
  of 
  the 
  double 
  curves 
  behave 
  in 
  the 
  

  

  envelope, 
  for 
  instance, 
  the 
  point 
  <£ 
  = 
  0, 
  t 
  = 
  0. 
  We 
  have 
  for 
  

  

  this 
  point 
  : 
  

  

  

  (-?)■ 
  

  

  