﻿concerning 
  Forced 
  Vibrations 
  and 
  Resonance. 
  105 
  

  

  The 
  existence 
  of 
  further 
  degrees 
  of 
  freedom 
  than 
  those 
  

   corresponding 
  to 
  the 
  given 
  force 
  M^ 
  and 
  the 
  disposable 
  force 
  

   ^ 
  makes 
  no 
  difference 
  to 
  (39). 
  And 
  so 
  long 
  as 
  M^? 
  % 
  are 
  

   the 
  only 
  forces 
  in 
  operation, 
  we 
  have 
  still 
  

  

  Mi2 
  2 
  Mod 
  2> 
  F 
  2 
  ,,, 
  N 
  

  

  Max. 
  work 
  = 
  — 
  ^— 
  ^ 
  — 
  : 
  ? 
  (41) 
  

  

  fc 
  A 
  n 
  sm 
  <x 
  n 
  

  

  If 
  further 
  all 
  the 
  coefficients 
  b 
  vanish, 
  except 
  b 
  22 
  , 
  

   (-10) 
  remains 
  unaffected. 
  If, 
  however, 
  we 
  suppose 
  that 
  b 
  m 
  

   b 
  u 
  , 
  etc. 
  are 
  finite, 
  while 
  b 
  n 
  , 
  h 
  l2 
  , 
  b^, 
  b 
  2 
  s, 
  &c. 
  still 
  vanish, 
  

   (20) 
  gives 
  

  

  — 
  p~ 
  l 
  A 
  n 
  sin 
  a 
  n 
  = 
  b 
  22 
  A 
  2l 
  2 
  + 
  b 
  3Z 
  A 
  n 
  2 
  + 
  . 
  . 
  ., 
  . 
  . 
  (1-2) 
  

   and 
  the 
  expression 
  for 
  the 
  maximum 
  work 
  becomes 
  

  

  i 
  A 
  2 
  Mnrl 
  2 
  *I> 
  

  

  g*M2 
  iU0Q 
  ^2 
  /^\ 
  

  

  b 
  22 
  A 
  12 
  w 
  -f- 
  6 
  3s 
  Ai3 
  + 
  . 
  . 
  . 
  

  

  Since 
  6 
  33 
  , 
  &c. 
  are 
  positive, 
  the 
  value 
  of 
  (43) 
  is 
  less 
  than 
  when 
  

   £ 
  ;53 
  , 
  &c. 
  vanish. 
  

  

  The 
  expression 
  (43) 
  is 
  necessarily 
  more 
  complicated 
  than 
  

   (40) 
  ; 
  but 
  a 
  simple 
  result 
  may 
  again 
  be 
  stated 
  if 
  we 
  suppose 
  

   that 
  given 
  forces 
  act 
  successively 
  of 
  the 
  second, 
  third, 
  and 
  

   following 
  types, 
  provided 
  they 
  be 
  of 
  such 
  magnitudes 
  that 
  

   they 
  would 
  severally 
  (the 
  non- 
  corresponding 
  resistances 
  

   vanishing) 
  allow 
  the 
  same 
  work 
  to 
  be 
  abstracted 
  by 
  ^Fj, 
  that 
  

   is 
  provided 
  

  

  Mod 
  2 
  ¥ 
  2 
  = 
  Mod 
  2 
  ^ 
  3 
  = 
  _ 
  Mod 
  2 
  ¥ 
  

  

  &22 
  ^33 
  f 
  > 
  

  

  (44) 
  

  

  On 
  this 
  supposition 
  the 
  sum 
  of 
  the 
  energies 
  abstractable 
  in 
  

   the 
  various 
  cases 
  has 
  the 
  value 
  

  

  Mod*¥ 
  

  

  ngg-> 
  ( 
  4 
  °) 
  

  

  of 
  the 
  same 
  form 
  as 
  before. 
  

  

  In 
  the 
  electrical 
  application 
  we 
  have 
  to 
  consider 
  any 
  

   number 
  of 
  mutually 
  influencing 
  circuits, 
  of 
  which 
  the 
  first 
  

   is 
  devoid 
  of 
  resistance. 
  The 
  electromotive 
  forces 
  acting 
  

   successively 
  in 
  the 
  other 
  circuits 
  are 
  to 
  be 
  inversely 
  as 
  the 
  

   square 
  roots 
  of 
  the 
  resistances 
  of 
  those 
  circuits, 
  i. 
  e. 
  such 
  as 
  

   would 
  do 
  the 
  same 
  amount 
  of 
  work 
  on 
  each 
  circuit 
  supposed 
  

   to 
  be 
  isolated 
  and 
  reduced 
  {e.g. 
  by 
  suitable 
  adjustment 
  of 
  the 
  

   associated 
  condenser) 
  to 
  a 
  mere 
  resistance. 
  The 
  sum 
  of 
  all 
  

   the 
  works 
  abstractable 
  in 
  the 
  first 
  circuit 
  is 
  then 
  the 
  same 
  as 
  

   if 
  there 
  were 
  no 
  other 
  circuits 
  than 
  the 
  first 
  and 
  second; 
  or, 
  

   again, 
  as 
  if 
  the 
  second 
  circuit 
  were 
  isolated 
  and 
  it 
  were 
  

   allowed 
  to 
  draw 
  work 
  from 
  it. 
  

  

  