﻿98 
  Lord 
  Rayleigh 
  on 
  some 
  General 
  Theorems 
  

  

  where 
  e 
  rs 
  denotes 
  the 
  quadratic 
  operator 
  

  

  e 
  ra 
  = 
  a 
  r8 
  D 
  2 
  + 
  b 
  rs 
  D 
  + 
  c 
  rs 
  (4) 
  

  

  And 
  it 
  is 
  to 
  be 
  remarked 
  that 
  since 
  

  

  it 
  follows 
  that 
  

  

  e 
  rs 
  = 
  eg? 
  yo) 
  

  

  If 
  we 
  multiply 
  the 
  first 
  of 
  equations 
  (3) 
  by 
  yjr 
  lf 
  the 
  second 
  

   by 
  *^ 
  2 
  > 
  & 
  c 
  »5 
  an 
  d 
  then 
  add, 
  we 
  obtain 
  

  

  ^±Z 
  ) 
  +2F 
  = 
  ^ 
  1 
  + 
  %f 
  2 
  + 
  (6) 
  

  

  In 
  this 
  the 
  first 
  term 
  represents 
  the 
  rate 
  at 
  which 
  energy 
  

   is 
  being 
  stored 
  in 
  the 
  system 
  ; 
  2F 
  is 
  the 
  rate 
  of 
  dissipation 
  ; 
  

   and 
  the 
  two 
  together 
  account 
  for 
  the 
  work 
  done 
  upon 
  the 
  

   system 
  in 
  time 
  dt 
  by 
  the 
  external 
  forces 
  ^ 
  l9 
  NPg, 
  &c. 
  

  

  In 
  considering 
  forced 
  vibrations 
  of 
  simple 
  type 
  we 
  take 
  

  

  ^! 
  = 
  E!^, 
  ¥ 
  2 
  =E 
  2 
  ^', 
  &c, 
  . 
  . 
  (7) 
  

  

  and 
  assume 
  that 
  yjr^ 
  y]r 
  2y 
  &c, 
  are 
  also 
  proportional 
  to 
  eP*. 
  

   The 
  coordinates 
  are 
  then 
  determined 
  by 
  the 
  system 
  of 
  

   algebraic 
  equations 
  resulting 
  from 
  the 
  substitution 
  in 
  (4), 
  

   (3) 
  of 
  ip 
  for 
  D. 
  The 
  most 
  general 
  motion 
  possible 
  under 
  

   the 
  assumed 
  forces 
  would 
  require 
  the 
  inclusion 
  of 
  free 
  vibra- 
  

   tions, 
  but 
  (unless 
  F 
  = 
  0) 
  these 
  die 
  out 
  as 
  time 
  progresses. 
  

  

  By 
  the 
  theory 
  of 
  determinants 
  the 
  solution 
  of 
  equations 
  (3) 
  

   may 
  be 
  expressed 
  in 
  the 
  form 
  

  

  where 
  V 
  denotes 
  the 
  determinant 
  of 
  the 
  symbols 
  e. 
  If 
  there 
  

   be 
  no 
  dissipation, 
  V, 
  or 
  as 
  we 
  may 
  write 
  it 
  with 
  fuller 
  ex- 
  

   pressiveness 
  V(«p) 
  , 
  is 
  an 
  even 
  function 
  of 
  ip 
  vanishing 
  when 
  

   p 
  corresponds 
  to 
  one 
  of 
  the 
  natural 
  frequencies 
  of 
  vibration. 
  

   In 
  such 
  a 
  case 
  the 
  coordinates 
  ^r 
  l5 
  &c, 
  in 
  general 
  become 
  

   infinite. 
  When 
  there 
  is 
  dissipation, 
  V(*/?) 
  does 
  not 
  vanish 
  

   for 
  any 
  (real) 
  value 
  of 
  p. 
  If 
  we 
  write 
  

  

  V(v)"=V,(v)+»>Vi(i», 
  .... 
  (9) 
  

  

  