﻿294 
  Mr. 
  H. 
  J. 
  E. 
  Beth 
  on 
  the 
  Oscillations 
  

  

  shall 
  call 
  the 
  osculating 
  curve 
  for 
  the 
  moment 
  indicated, 
  

   which 
  name 
  is 
  in 
  use 
  in 
  the 
  theory 
  of 
  disturbances 
  (see, 
  

   among 
  others, 
  H. 
  Poincare, 
  Lemons 
  tie 
  Mecanigue 
  celeste, 
  vol. 
  i. 
  

   p. 
  90). 
  As 
  f, 
  fix, 
  and 
  /3 
  2 
  vary 
  slowly, 
  the 
  osculating 
  curve 
  

   will 
  also 
  vary 
  ; 
  we 
  shall 
  examine 
  the 
  system 
  of 
  Lissajous 
  

   curves 
  described. 
  

  

  We 
  take 
  a 
  new 
  unit 
  of 
  length, 
  having 
  Ji 
  h 
  old 
  units 
  ; 
  

  

  then 
  the 
  amplitudes 
  become 
  resp. 
  ^/fand 
  -y/l 
  — 
  £. 
  We 
  see 
  

  

  immediately 
  that 
  the 
  vertices 
  of 
  the 
  rectangles 
  in 
  which 
  the 
  

   Lissajous 
  curves 
  are 
  described 
  lie 
  on 
  the 
  circumference 
  of 
  

   an 
  ellipse 
  with 
  its 
  great 
  axis 
  along 
  the 
  X-axis 
  ; 
  the 
  half 
  of 
  

   the 
  great 
  axis 
  is 
  the 
  new 
  unit 
  of 
  length 
  ; 
  the 
  half 
  of 
  the 
  

   small 
  axis 
  is 
  the 
  7th 
  part 
  of 
  the 
  new 
  unit 
  of 
  length. 
  

  

  We 
  may 
  for 
  each 
  osculating 
  curve 
  choose 
  the 
  origin 
  of 
  

   time 
  in 
  such 
  a 
  way 
  that 
  for 
  t 
  = 
  the 
  moving 
  point 
  has 
  its 
  

   greatest 
  deviation 
  to 
  the 
  right. 
  The 
  difference 
  in 
  phase 
  is 
  

   2yn 
  x 
  (f3 
  x 
  — 
  fiy) 
  = 
  <f>, 
  when 
  the 
  phase 
  is 
  calculated 
  from 
  this 
  

   same 
  moment. 
  Taking 
  at 
  last 
  a 
  new 
  unit 
  of 
  time, 
  we 
  may 
  

   write 
  the 
  equations 
  in 
  this 
  form 
  : 
  

  

  %= 
  a/? 
  cos 
  t 
  

   1 
  

  

  y=- 
  v'l 
  — 
  £cos 
  (yt—cj>) 
  

  

  (17) 
  

  

  Every 
  osculating 
  curve 
  is 
  described 
  in 
  2ir 
  new 
  units 
  of 
  

   time. 
  

  

  The 
  relation 
  between 
  f 
  and 
  cf> 
  has 
  the 
  form 
  : 
  

  

  rl(i~rfcos^,=/(r), 
  .... 
  (is) 
  

  

  where 
  /(?) 
  has 
  for 
  each 
  of 
  the 
  cases 
  7=1 
  ($ 
  1 
  = 
  2), 
  y 
  = 
  2 
  

   ■(Si 
  = 
  3j, 
  and 
  7=3 
  (S! 
  = 
  4) 
  another 
  form, 
  namely 
  : 
  

  

  7 
  = 
  2 
  f(£)szk+p% 
  

   7-3 
  /(£)=pg«- 
  + 
  tf 
  +f 
  .. 
  

  

  7=1 
  / 
  (?) 
  = 
  - 
  « 
  \l 
  ± 
  v>? 
  2 
  + 
  <fc 
  + 
  r 
  + 
  \l\ 
  

  

  § 
  28. 
  The 
  shape 
  of 
  a 
  Lissajous 
  curve 
  depends 
  on 
  the 
  ratio 
  

   of 
  the 
  periods, 
  the 
  ratio 
  of 
  the 
  amplitudes, 
  and 
  the 
  difference 
  

   in 
  phase. 
  The 
  shape 
  of 
  the 
  curves 
  for 
  the 
  ratio 
  -]-, 
  ^, 
  \ 
  of 
  

   the 
  periods 
  is 
  well 
  known 
  for 
  arbitrary 
  values 
  of 
  <£ 
  as 
  well 
  

   as 
  for 
  the 
  particular 
  values 
  of 
  <£ 
  for 
  which 
  sin 
  <£ 
  = 
  or 
  

   cos 
  </> 
  = 
  0. 
  For 
  sin 
  (fi 
  = 
  we 
  obtain 
  the 
  curves 
  which 
  are 
  

   described 
  in 
  both 
  directions 
  alternately 
  ; 
  we 
  shall 
  call 
  them 
  

   Lissajous 
  double 
  curves. 
  That 
  the 
  condition 
  sin 
  </> 
  = 
  is 
  

  

  

  