﻿concerning 
  Forced 
  Vibrations 
  and 
  Resonance. 
  101 
  

  

  vibrations 
  are 
  isoperiodic 
  with 
  the 
  forced 
  vibrations. 
  The 
  

   maximum 
  value 
  itself 
  is 
  

  

  iM 
  f- 
  (*) 
  

  

  Let 
  us 
  now 
  suppose 
  that 
  two 
  forces 
  act 
  upon 
  the 
  system, 
  

   one 
  of 
  which 
  M* 
  is 
  given, 
  while 
  the 
  second 
  "SP' 
  is 
  at 
  disposal, 
  

   and 
  let 
  us 
  inquire 
  how 
  much 
  work 
  can 
  be 
  withdrawn 
  by 
  ^' 
  . 
  

   It 
  will 
  probably 
  conduce 
  to 
  clearness 
  if 
  we 
  think 
  of 
  an 
  

   electric 
  circuit 
  possessing 
  self-induction 
  and 
  resistance, 
  and 
  

   closed 
  by 
  a 
  condenser, 
  so 
  as 
  to 
  constitute 
  a 
  vibrator. 
  In 
  this 
  

   acts 
  a 
  given 
  electromotive 
  force 
  ^ 
  of 
  given 
  frequency. 
  At 
  

   another 
  part 
  of 
  the 
  circuit 
  another 
  electromotive 
  force 
  can 
  be 
  

   introduced, 
  and 
  the 
  question 
  is 
  what 
  work 
  can 
  be 
  obtained 
  at 
  

   that 
  point. 
  Of 
  course 
  any 
  work 
  so 
  obtained, 
  as 
  well 
  as 
  that 
  

   dissipated 
  in 
  the 
  system, 
  must 
  be 
  introduced 
  by 
  the 
  operation 
  

   of 
  the 
  given 
  force 
  M*. 
  

  

  It 
  will 
  suffice 
  for 
  the 
  moment 
  to 
  take 
  ^ 
  such 
  that 
  yjr 
  due 
  

   to 
  it 
  is 
  unity, 
  which 
  will 
  happen 
  when 
  ^ 
  = 
  A~ 
  l 
  e~ 
  ia 
  . 
  If 
  M*' 
  be 
  

   Re 
  9 
  , 
  the 
  complete 
  value 
  of 
  yjr 
  is 
  

  

  t|t 
  = 
  l 
  + 
  AR^ 
  a 
  +0 
  } 
  (27) 
  

  

  The 
  work 
  done 
  (in 
  unit 
  time) 
  by 
  ^' 
  consists 
  of 
  two 
  parts. 
  

   That 
  corresponding 
  to 
  the 
  second 
  term 
  in 
  (27) 
  is 
  the 
  same 
  as 
  

   if 
  M*' 
  had 
  acted 
  alone 
  and, 
  as 
  in 
  (14), 
  its 
  value 
  is 
  

  

  — 
  i#R 
  2 
  Asina. 
  

  

  The 
  work 
  done 
  by 
  W 
  upon 
  the 
  first 
  part 
  of 
  i|r 
  given 
  in 
  (27) 
  

   is, 
  as 
  in 
  (22), 
  

  

  -ipEsin(-0). 
  

  

  The 
  whole 
  work 
  done 
  by 
  ^ 
  is 
  found 
  by 
  adding 
  these 
  

   together 
  ; 
  and 
  the 
  work 
  withdrawn 
  from 
  the 
  system 
  by 
  M/ 
  7 
  is 
  

   the 
  negative 
  of 
  this, 
  or 
  

  

  ±pWA*ma 
  — 
  ±plismd 
  (28) 
  

  

  In 
  this 
  expression 
  the 
  first 
  term 
  is 
  negative, 
  and 
  the 
  whole 
  is 
  

   to 
  be 
  made 
  a 
  maximum 
  by 
  variation 
  of 
  R 
  and 
  0. 
  The 
  maxi- 
  

   mum 
  occurs 
  when 
  

  

  sin0= 
  — 
  1, 
  2KAsin*=— 
  1; 
  . 
  . 
  (29) 
  

  

  and 
  the 
  maximum 
  value 
  itself 
  is 
  

  

  8A 
  sin 
  a. 
  

  

  (30) 
  

  

  