﻿Nongenerating 
  Force 
  of 
  High 
  Frequency. 
  621 
  

  

  Bach 
  of 
  the 
  equations 
  (0)' 
  therefore 
  gives 
  

  

  oo 
  « 
  2 
  . 
  O 
  2 
  

  

  A-r 
  + 
  Z/ri 
  , 
  =<J, 
  

  

  i. 
  6. 
  

  

  00 
  

  

  Now 
  $ 
  {f(i)+f(—i)}dt 
  = 
  2n 
  2 
  '£^ 
  ot 
  r 
  cosrntdt 
  

  

  1 
  

  

  = 
  2/i 
  2 
  -- 
  smr#£, 
  

   i 
  r 
  

  

  so 
  that 
  2>i 
  2 
  2( 
  — 
  ) 
  is 
  the 
  mean 
  value 
  of 
  the 
  function 
  

   i 
  \ 
  r 
  i 
  

  

  [J 
  o 
  {/(0+/(-0}*]\ 
  

  

  Again, 
  

  

  J 
  .{/(*)-/.( 
  " 
  0}* 
  = 
  2^ 
  2 
  f 
  J 
  A 
  sin 
  ™* 
  <** 
  

  

  = 
  2n2-- 
  (1 
  — 
  cos 
  r?i£). 
  

  

  i 
  r 
  v 
  y 
  

  

  Therefore 
  if 
  M 
  denotes 
  the 
  mean 
  value 
  of 
  the 
  integral 
  over 
  

   a 
  period 
  

  

  2n2^cosm*=M-f 
  {f(t)-f(-t)\dt; 
  

  

  i 
  r 
  ° 
  

  

  whence 
  it 
  follows 
  that 
  2n 
  2 
  %[—\ 
  is 
  the 
  mean 
  value 
  of 
  

  

  [«-J.y(0-/<-0Kr- 
  ' 
  

  

  Thus 
  if 
  the 
  spring 
  of 
  an 
  oscillation 
  consists 
  of 
  any 
  periodic 
  

   variation 
  2f{t) 
  about 
  a 
  mean 
  value, 
  \, 
  which 
  is 
  small 
  in 
  

   comparison 
  with 
  n 
  2 
  , 
  the 
  square 
  of 
  the 
  variation 
  frequency, 
  

   and 
  if, 
  furthermore, 
  the 
  ratio 
  f(t)/n 
  2 
  is 
  always 
  small, 
  then 
  the 
  

   resulting 
  mean 
  motion 
  is 
  simple 
  oscillatory 
  of 
  spring 
  equal 
  

   to 
  X 
  together 
  with 
  the 
  sum 
  of 
  the 
  mean 
  values 
  of 
  the 
  periodic 
  

   functions 
  

  

  USf(f)+A-t)}dtJ 
  

  

  where 
  M 
  is 
  the 
  mean 
  value 
  of 
  f 
  {/(0 
  -/( 
  — 
  t)}dt. 
  

  

  Manchester, 
  

  

  April, 
  1908. 
  

  

  