﻿100 
  Lord 
  Rayleigh 
  on 
  some 
  General 
  Theorems 
  

  

  so 
  that 
  by 
  (15) 
  the 
  work 
  dissipated 
  in 
  time 
  t 
  is 
  

  

  ip 
  2 
  R 
  1 
  2 
  t\b 
  11 
  A 
  11 
  2 
  + 
  b 
  22 
  A 
  2l 
  2 
  + 
  . 
  .. 
  

  

  + 
  2/>i 
  2 
  AnA 
  12 
  cos 
  (an 
  — 
  a 
  12 
  ) 
  + 
  . 
  • 
  •}• 
  • 
  (19) 
  

  

  Equating 
  the 
  equivalent 
  quantities 
  in 
  (14) 
  and 
  (19), 
  we 
  get 
  

  

  — 
  ^ 
  _1 
  A 
  1 
  isina 
  1 
  i 
  = 
  ?>nA 
  1 
  i 
  -f 
  /> 
  22 
  A 
  21 
  + 
  . 
  . 
  . 
  

  

  + 
  2/> 
  12 
  A 
  1 
  iAi 
  2 
  cos(a 
  11 
  — 
  a 
  12 
  ) 
  + 
  (20) 
  

  

  This 
  assumes 
  a 
  specially 
  simple 
  form 
  when 
  F 
  is 
  a 
  function 
  of 
  

   the 
  squares 
  only 
  of 
  d^/dt, 
  &c, 
  so 
  that 
  6 
  12 
  , 
  &c. 
  vanish. 
  

  

  In 
  (14) 
  we 
  have 
  calculated 
  the 
  work 
  done 
  by 
  a 
  force 
  M^ 
  

   acting 
  alone 
  upon 
  the 
  system. 
  If 
  other 
  forces 
  act, 
  the 
  ex- 
  

   pression 
  for 
  a/tj 
  will 
  deviate 
  from 
  (13); 
  but 
  in 
  any 
  case 
  we 
  

   may 
  write 
  

  

  ^ 
  = 
  R/\ 
  ^ 
  1==V 
  «V. 
  • 
  • 
  (21) 
  

  

  and 
  the 
  work 
  done 
  in 
  unit 
  of 
  time 
  by 
  the 
  real 
  part 
  of 
  ^J 
  r 
  1 
  on 
  

   the 
  real 
  part 
  of 
  ^ 
  will 
  be 
  

  

  -ipRrsin^-^), 
  (22) 
  

  

  and 
  depends 
  upon 
  the 
  product 
  of 
  the 
  moduli 
  and 
  the 
  difference 
  

   of 
  phases. 
  

  

  If 
  ^i 
  consist 
  of 
  two 
  or 
  more 
  parts 
  of 
  the 
  form 
  (21), 
  the 
  

   work 
  done 
  is 
  to 
  be 
  found 
  by 
  addition 
  of 
  the 
  terms 
  corre- 
  

   sponding 
  to 
  the 
  various 
  parts. 
  

  

  One 
  Degree 
  of 
  Freedom. 
  

  

  The 
  theory 
  of 
  the 
  vibrations 
  of 
  a 
  system 
  of 
  one 
  degree 
  of 
  

   freedom, 
  resulting 
  from 
  the 
  application 
  of 
  a 
  given 
  force, 
  is 
  

   simple 
  and 
  well 
  known, 
  but 
  it 
  will 
  be 
  convenient 
  to 
  make 
  a 
  

   few 
  remarks 
  and 
  deductions. 
  

  

  The 
  equation 
  determining 
  yjr 
  in 
  terms 
  of 
  'M* 
  is 
  

  

  (-apt+d 
  + 
  ipfyifr-W; 
  .... 
  (23) 
  

  

  so 
  that 
  in 
  the 
  notation 
  of 
  (10) 
  

  

  Ae^ 
  = 
  j|— 
  (24) 
  

  

  c 
  — 
  apr 
  + 
  ipl> 
  ' 
  

  

  As 
  in 
  (14), 
  the 
  work 
  done 
  by 
  the 
  force 
  in 
  unit 
  time 
  is 
  

   Jp^bMod^ 
  

   (c-ap 
  2 
  ) 
  2 
  +p 
  2 
  ~b 
  2 
  ' 
  [ 
  } 
  

  

  and 
  it 
  reaches 
  a 
  maximum 
  (1> 
  and 
  p 
  being 
  given) 
  when 
  the 
  

   tuning 
  is 
  such 
  that 
  c— 
  ap 
  2 
  = 
  0, 
  that 
  is 
  when 
  the 
  natural 
  

  

  

  