﻿concerning 
  Forced 
  Vibrations 
  and 
  Resonance, 
  99 
  

  

  in 
  which 
  Vi, 
  V2 
  are 
  even 
  functions 
  of 
  ip, 
  V2 
  depends 
  entirely 
  

   upon 
  the 
  dissipation, 
  while 
  if 
  the 
  dissipation 
  be 
  small, 
  Vi 
  is 
  

   approximately 
  the 
  same 
  as 
  if 
  there 
  were 
  none. 
  

  

  As 
  it 
  will 
  be 
  convenient 
  to 
  have 
  a 
  briefer 
  notation 
  than 
  

   that 
  of 
  (8), 
  we 
  will 
  write 
  

  

  ^ 
  2 
  = 
  A 
  21 
  ^¥H-A 
  22 
  ^ 
  22 
  %+. 
  .. 
  > 
  . 
  . 
  (10) 
  

  

  in 
  which 
  A, 
  a 
  are 
  real 
  and 
  are 
  subject 
  to 
  the 
  relations 
  

  

  A 
  rs 
  = 
  A 
  s 
  ,,, 
  a 
  rs 
  = 
  a 
  8r 
  (11) 
  

  

  In 
  order 
  to 
  take 
  account 
  of 
  the 
  phases 
  of 
  the 
  forces, 
  we 
  may 
  

   suppose 
  similarly 
  that 
  in 
  (7) 
  

  

  E 
  1 
  = 
  -rV^ 
  E 
  2 
  = 
  tV 
  02 
  , 
  &c. 
  . 
  . 
  (12) 
  

  

  Work 
  Done. 
  

  

  If 
  we 
  suppose 
  that 
  but 
  one 
  force, 
  say 
  M^, 
  acts 
  upon 
  the 
  

   system, 
  the 
  values 
  of 
  the 
  coordinates 
  are 
  given 
  by 
  the 
  first 
  

   terms 
  of 
  the 
  right-hand 
  members 
  of 
  (10). 
  The 
  work 
  done 
  

   by 
  the 
  force 
  in 
  time 
  dt 
  depends 
  upon 
  that 
  part 
  of 
  dfa/dt 
  

   which 
  is 
  in 
  the 
  same 
  phase 
  with 
  it, 
  corresponding 
  to 
  the 
  part 
  

   of 
  ^ 
  which 
  is 
  in 
  quadrature 
  with 
  the 
  force. 
  Thus, 
  taking 
  

   the 
  real 
  parts 
  only 
  of 
  the 
  symbolic 
  quantities, 
  so 
  that 
  

  

  ^ 
  1 
  = 
  R 
  1 
  cos(^ 
  + 
  l9 
  l 
  ) 
  ? 
  ^ 
  1= 
  AnBi 
  cos 
  (^ 
  + 
  #! 
  + 
  «!!), 
  . 
  (13) 
  

   we 
  have 
  as 
  the 
  work 
  done 
  (on 
  the 
  average) 
  in 
  time 
  t 
  

   — 
  pAnKi 
  2 
  [cos 
  {pt 
  + 
  0J 
  . 
  sin 
  (pt 
  + 
  6 
  X 
  + 
  « 
  u 
  ) 
  dt, 
  

   or 
  

  

  -ipRi* 
  A 
  u 
  sin 
  a 
  n 
  . 
  t 
  (14; 
  

  

  As 
  was 
  to 
  be 
  expected, 
  this 
  is 
  independent 
  of 
  V 
  

  

  Another 
  expression 
  for 
  the 
  same 
  quantity 
  may 
  be 
  obtained 
  

   by 
  considering 
  how 
  this 
  work 
  is 
  dissipated. 
  From 
  (6) 
  we 
  

   see 
  that 
  

  

  = 
  b 
  n 
  $iridt 
  + 
  b 
  22 
  §f2 
  2 
  dt 
  + 
  . 
  . 
  . 
  + 
  26 
  u 
  j'^if»<ft 
  + 
  . 
  . 
  . 
  . 
  (15) 
  

   Taking 
  again 
  the 
  real 
  parts 
  in 
  (10), 
  we 
  have 
  

  

  $jr; 
  2 
  dt 
  = 
  y 
  2 
  R 
  l 
  2 
  A 
  n 
  2 
  .t. 
  (16; 
  

  

  J^l^ 
  , 
  2^-=i/Ri 
  2 
  AiiAi2C0s(a 
  11 
  -rai 
  2 
  ).^ 
  . 
  . 
  (17) 
  

  

  ^■,yjr-,dt 
  = 
  ip 
  2 
  R 
  l 
  2 
  A 
  l2 
  A 
  is 
  cos 
  (a 
  12 
  -«i 
  3 
  ) 
  . 
  t 
  ; 
  (18) 
  

  

  H2 
  

  

  