﻿On 
  Forced 
  Vibrations 
  and 
  Resonance. 
  97 
  

  

  shown 
  iu 
  (2). 
  In 
  the 
  course 
  of 
  this 
  process 
  we 
  have 
  

  

  dw 
  = 
  Fdf+ 
  Y'df' 
  + 
  Qdg 
  4- 
  Q'dg' 
  + 
  Udh 
  + 
  RW 
  ... 
  (3) 
  . 
  

  

  Hence 
  if 
  we 
  suppose 
  10 
  expressed 
  as 
  a 
  function 
  of 
  /, 
  /', 
  

   g, 
  g', 
  h, 
  h', 
  we 
  have 
  

  

  die 
  -p 
  dw 
  -p, 
  dw 
  _ 
  n 
  c?tu 
  ~, 
  dw 
  ' 
  j} 
  dw 
  t>/ 
  . 
  

  

  <y- 
  F 
  ' 
  <r 
  ^ 
  _Q 
  ' 
  <¥ 
  =Q 
  ' 
  * 
  ~ 
  ' 
  5ft' 
  =R 
  • 
  • 
  • 
  ( 
  ; 
  - 
  

  

  This 
  completes 
  the 
  foundation 
  of 
  the 
  molar 
  dynamics 
  of 
  an 
  

   elastic 
  solid 
  of 
  the 
  most 
  general 
  possible 
  kind 
  according 
  to 
  

   Green's 
  theory, 
  expressed 
  in 
  terms 
  of 
  the 
  new 
  mode 
  of 
  

   specifying 
  stresses 
  and 
  strains. 
  

  

  In 
  a 
  communication 
  to 
  the 
  Royal 
  Society 
  of 
  Edinburgh 
  

   promised 
  for 
  Jan. 
  20, 
  1902, 
  I 
  hope 
  to 
  use 
  with 
  advantage 
  this 
  

   mode 
  of 
  specification 
  in 
  working 
  out 
  some 
  details 
  of 
  the 
  

   molecular 
  dynamics 
  of 
  a 
  crystal. 
  

  

  . 
  _ 
  _ 
  x 
  

  

  IX. 
  Some 
  General 
  Theorems 
  concerning 
  Forced 
  Vibrations 
  

   and 
  Resonance. 
  By 
  Lord 
  Rayleigh, 
  F.R.S* 
  

  

  THE 
  general 
  equation 
  for 
  the 
  small 
  vibrations 
  of 
  a 
  system 
  

   whose 
  configuration 
  is 
  defined 
  by 
  the 
  generalized 
  co- 
  

   ordinates 
  fa, 
  fa, 
  ... 
  . 
  may 
  be 
  written 
  t 
  

  

  *** 
  g 
  

  

  dtdyjr 
  d^r 
  d^r 
  v 
  ' 
  

  

  where 
  T, 
  F, 
  V, 
  denoting 
  respectively 
  the 
  kinetic 
  energy, 
  the 
  

   dissipation 
  function, 
  and 
  the 
  potential 
  energy, 
  have 
  the 
  forms 
  

  

  T 
  = 
  &t 
  n 
  fa 
  2 
  + 
  ^22^2 
  2 
  + 
  . 
  . 
  . 
  + 
  d 
  l2 
  fafa 
  + 
  ■ 
  • 
  • 
  \ 
  

  

  F 
  = 
  ibnff 
  + 
  ib 
  22 
  fa* 
  + 
  . 
  . 
  . 
  + 
  b 
  12 
  fafa 
  + 
  ... 
  L 
  . 
  (2) 
  

  

  V 
  = 
  ic 
  n 
  fa 
  2 
  + 
  \c 
  22 
  fa 
  2 
  + 
  ...*+ 
  G 
  12 
  fafa 
  + 
  . 
  • 
  • 
  J 
  

  

  in 
  which 
  the 
  coefficients 
  a, 
  b, 
  c 
  are 
  constants. 
  

  

  If 
  we 
  substitute 
  in 
  (1) 
  the 
  values 
  of 
  T, 
  F, 
  and 
  V, 
  and 
  write 
  

   D 
  for 
  d/dt, 
  we 
  obtain 
  a 
  system 
  of 
  equations 
  which 
  may 
  be 
  

   put 
  into 
  the 
  form 
  

  

  e 
  n 
  fa 
  + 
  e 
  X2 
  fa 
  + 
  e^fa 
  4- 
  . 
  . 
  . 
  = 
  ^1 
  

  

  ?2lfa 
  + 
  ^22^2 
  + 
  *23^3 
  + 
  • 
  - 
  • 
  = 
  "^2 
  l 
  (;>) 
  

  

  ^31^1 
  + 
  ^32'f 
  2 
  + 
  enfa 
  + 
  •'.. 
  = 
  ^ 
  a 
  

  

  * 
  Communicated 
  by 
  tlie 
  Author. 
  

  

  f 
  See 
  ' 
  Theory 
  of 
  Sound,' 
  vol. 
  1. 
  §§ 
  82, 
  84. 
  104. 
  

  

  Phil. 
  Mag. 
  S. 
  6. 
  Vol. 
  3. 
  No. 
  13. 
  Jan. 
  1902, 
  H 
  

  

  