﻿Periodic 
  Nongenerating 
  Force 
  of 
  High 
  Frequency. 
  617 
  

  

  by 
  Fourier's 
  theorem, 
  and 
  hence 
  c 
  obtained 
  ; 
  but 
  there 
  is 
  

   a 
  much 
  simpler 
  method 
  of 
  procedure 
  applicable 
  alike 
  to 
  

   general 
  and 
  numerical 
  cases. 
  

  

  If 
  A 
  denotes 
  the 
  variable 
  part 
  of 
  the 
  spring 
  about 
  the 
  

   mean 
  value, 
  X, 
  then 
  

  

  §Adt 
  =27i 
  2 
  2j 
  o 
  a,cos 
  

  

  mt 
  dt 
  

  

  =r 
  2n 
  ^ 
  sin 
  mt 
  

  

  1 
  r 
  

  

  Hence 
  

  

  J 
  f 
  A 
  dt 
  } 
  2 
  = 
  2n 
  2 
  2 
  (- 
  ) 
  + 
  4» 
  2 
  2 
  jr 
  cos 
  mt. 
  

  

  Thus 
  2ri 
  2 
  X 
  ( 
  — 
  ) 
  is 
  the 
  mean 
  value 
  of 
  the 
  square 
  of 
  the 
  

  

  time 
  integral, 
  from 
  * 
  = 
  0, 
  of 
  the 
  spring 
  variation 
  A. 
  

  

  If, 
  therefore, 
  the 
  spring 
  of 
  a 
  system 
  is 
  an 
  even 
  function 
  of 
  

   the 
  time, 
  of 
  high 
  frequency, 
  and 
  if 
  the 
  ratio 
  of 
  the 
  amplitude 
  

   of 
  each 
  elementary 
  variation 
  to 
  the 
  square 
  of 
  the 
  frequency 
  

   is 
  small, 
  then 
  the 
  mean 
  motion 
  is 
  the 
  same 
  as 
  if 
  the 
  spring 
  

   were 
  constant 
  and 
  equal 
  to 
  the 
  mean 
  spring 
  X, 
  together 
  with 
  

   the 
  mean 
  value 
  of 
  the 
  square 
  of 
  the 
  time 
  integral 
  of 
  spring 
  

   variation 
  about 
  X, 
  the 
  range 
  of 
  integration 
  being 
  a 
  period 
  so 
  

   chosen 
  that 
  the 
  values 
  of 
  the 
  spring 
  at 
  equal 
  intervals 
  from 
  

   its 
  limits 
  are 
  equal. 
  

  

  2. 
  As 
  an 
  illustration 
  let 
  us 
  take 
  the 
  extreme 
  case 
  of 
  

   impulsive 
  disturbance 
  which 
  has 
  already 
  been 
  examined 
  by 
  

   a 
  different 
  method 
  *. 
  

  

  If 
  the 
  variation 
  of 
  spring 
  is 
  made 
  up 
  of 
  equal 
  and 
  opposite 
  

   constant 
  changes 
  acting 
  alternately 
  for 
  equal 
  times 
  and 
  

   following 
  at 
  equal 
  intervals, 
  its 
  graph 
  over 
  a 
  period 
  is 
  

  

  C 
  

  

  F 
  

  

  ABCDEF 
  ... 
  as 
  shown 
  in 
  the 
  diagram, 
  the 
  period 
  being- 
  

   chosen 
  in 
  accordance 
  with 
  the 
  preceding 
  rule. 
  

  

  * 
  In 
  the 
  latter 
  of 
  the 
  papers 
  referred 
  to 
  above. 
  

  

  