﻿concerning 
  Forced 
  Vibrations 
  and 
  Resonance. 
  103 
  

  

  Several 
  Degrees 
  of 
  Freedom. 
  

  

  Hitherto 
  we 
  have 
  supposed 
  that 
  the 
  force 
  by 
  means 
  of 
  

   which 
  work 
  is 
  abstracted 
  is 
  of 
  the 
  same 
  type 
  as 
  that 
  which 
  is 
  

   supposed 
  to 
  be 
  given 
  and 
  by 
  means 
  of 
  which 
  work 
  is 
  intro- 
  

   duced 
  into 
  the 
  system 
  ; 
  but 
  in 
  the 
  investigations 
  which 
  

   follow 
  it 
  will 
  be 
  our 
  object 
  to 
  trace 
  the 
  effect 
  of 
  voluntary 
  

   operation 
  upon 
  one 
  coordinate 
  fa, 
  while 
  the 
  system 
  is 
  subject 
  

   to 
  given 
  forces 
  ^P^ 
  ^ 
  3 
  , 
  &c, 
  operating 
  upon 
  the 
  remaining 
  

   coordinates. 
  To 
  explain 
  what 
  is 
  meant 
  the 
  more 
  clearly, 
  let 
  

   us 
  consider 
  the 
  simple 
  case 
  of 
  two 
  electric 
  circuits 
  influencing 
  

   one 
  another 
  by 
  induction. 
  Each 
  circuit 
  may 
  be 
  supposed 
  to 
  

   be 
  closed 
  by 
  a 
  condenser, 
  so 
  as 
  to 
  constitute, 
  when 
  considered 
  

   by 
  itself, 
  a 
  simple 
  vibrator. 
  If 
  a 
  given 
  periodic 
  electro- 
  

   motive 
  force 
  ("^2) 
  act 
  in 
  the 
  second 
  circuit, 
  the 
  current 
  in 
  the 
  

   first 
  circuit 
  would 
  depend 
  upon 
  the 
  various 
  elements 
  of 
  the 
  

   compound 
  system. 
  Let 
  it 
  be 
  proposed 
  to 
  inquire 
  "what 
  work 
  

   can 
  be 
  withdrawn 
  at 
  the 
  first 
  circuit 
  by 
  electromotive 
  forces 
  

   (^Fi) 
  there 
  applied. 
  For 
  simplicity 
  it 
  will 
  be 
  supposed 
  that 
  

   the 
  first 
  circuit 
  has 
  no 
  resistance 
  (b 
  n 
  = 
  0). 
  

  

  When 
  ^ 
  2 
  acts 
  alone, 
  the 
  fa 
  due 
  to 
  it 
  is 
  given 
  by 
  

  

  ti=Ai,^ 
  (33) 
  

  

  If 
  we 
  ascribe 
  to 
  ^ 
  2 
  the 
  value 
  Ai 
  2 
  — 
  1 
  e~ 
  ta 
  > 
  2 
  , 
  the 
  ^1 
  due 
  to 
  it 
  will 
  

   be 
  unity, 
  and 
  this 
  for 
  the 
  present 
  purpose 
  is 
  the 
  simplest 
  

   supposition 
  to 
  make. 
  When 
  a 
  force 
  M*i 
  is 
  introduced, 
  the 
  

   complete 
  value 
  of 
  fa 
  will 
  deviate 
  from 
  unity, 
  but 
  ^ 
  2 
  is 
  

   supposed 
  to 
  retain 
  throughout 
  the 
  above 
  prescribed 
  value. 
  If 
  

   ^ 
  1 
  = 
  R 
  1 
  e* 
  01 
  , 
  as 
  in 
  (12), 
  the 
  complete 
  expression 
  of 
  fa 
  is 
  

  

  ^ 
  1 
  =l+A 
  11 
  R 
  1 
  ^i+*") 
  (34) 
  

  

  The 
  work 
  done 
  by 
  S 
  P 
  1 
  upon 
  this 
  is 
  composed 
  of 
  two 
  parts. 
  

   If 
  the 
  real 
  components 
  of 
  the 
  expressions 
  be 
  retained, 
  the 
  

   first 
  is 
  

  

  — 
  #f 
  R 
  x 
  cos 
  [pt 
  + 
  0J 
  . 
  sin 
  pt 
  dt=^pR 
  x 
  sin 
  6 
  X 
  . 
  t 
  ; 
  

  

  and 
  the 
  second 
  is, 
  as 
  in 
  (14), 
  

  

  —ipE, 
  a 
  A 
  u 
  sin 
  on.*, 
  

  

  a 
  quantity 
  necessarily 
  positive. 
  

  

  Thus 
  altogether 
  the 
  work 
  done 
  by 
  ^P, 
  in 
  unit 
  time 
  is 
  

  

  ^RiSin^-^R^Ansinan. 
  . 
  . 
  . 
  (35) 
  

  

  The 
  work 
  that 
  may 
  be 
  abstracted 
  from 
  the 
  circuit 
  is 
  the 
  

   negative 
  of 
  this, 
  and 
  it 
  is 
  to 
  be 
  made 
  a 
  maximum 
  by 
  variation 
  

   of 
  Rj 
  and 
  ft. 
  We 
  must 
  take 
  

  

  sin 
  ft 
  =-1, 
  2R 
  1 
  A 
  u 
  sin« 
  11 
  =— 
  1 
  ; 
  . 
  . 
  (36) 
  

  

  