﻿116 
  . 
  Lord 
  Rayleigh 
  on 
  some 
  General 
  Theorems 
  

  

  If 
  n 
  be 
  one 
  of 
  the 
  values 
  of 
  p 
  corresponding 
  to 
  maximum 
  

   resonance, 
  the 
  real 
  part 
  of 
  (87) 
  vanishes 
  when 
  p=n 
  ; 
  so 
  that 
  

  

  (c 
  u 
  — 
  n 
  2 
  a 
  n 
  )(c 
  22 
  — 
  n 
  2 
  a 
  22 
  ) 
  — 
  (c 
  l: 
  ,— 
  n 
  2 
  a 
  12 
  ) 
  2 
  = 
  0, 
  . 
  . 
  (88) 
  

  

  or 
  written 
  as 
  a 
  quadratic 
  in 
  ri\ 
  

  

  CnC 
  2 
  2— 
  c 
  12 
  2 
  — 
  n 
  2 
  (a 
  n 
  c 
  22 
  + 
  a 
  22 
  c 
  n 
  + 
  2a 
  l2 
  c 
  l2 
  ) 
  

  

  + 
  n 
  i 
  {a 
  n 
  a 
  22 
  -a, 
  2 
  2 
  )=0.. 
  . 
  . 
  . 
  (89) 
  

  

  By 
  subtraction 
  of 
  (89), 
  (87) 
  may 
  be 
  written 
  

  

  e 
  n 
  e 
  22 
  — 
  e 
  12 
  2 
  = 
  —(p 
  2 
  — 
  n 
  2 
  ) 
  (a 
  n 
  c 
  22 
  + 
  a 
  22 
  c 
  n 
  + 
  2a 
  12 
  c 
  12 
  ) 
  

  

  + 
  ( 
  p 
  ^-^)(a 
  n 
  a 
  22 
  -aJ)+ipbJc 
  u 
  ^p 
  2 
  a 
  ll 
  ). 
  . 
  . 
  (90) 
  

  

  If 
  b 
  22 
  were 
  zero, 
  t^ 
  would 
  become 
  infinite 
  for 
  p 
  = 
  n. 
  If 
  

   we 
  assume 
  that 
  b 
  22 
  , 
  while 
  not 
  actually 
  zero, 
  is 
  still 
  relatively 
  

   very 
  small, 
  the 
  values 
  of 
  p 
  in 
  the 
  neighbourhood 
  of 
  n 
  retain 
  

   a 
  preponderating 
  importance 
  ; 
  and 
  we 
  may 
  equate 
  p 
  to 
  n 
  with 
  

   exception 
  of 
  the 
  factor 
  (p 
  — 
  n). 
  Thus 
  (86), 
  (90) 
  become 
  

  

  e 
  i2 
  = 
  C 
  12 
  — 
  n2a 
  u1 
  ( 
  91 
  ) 
  

  

  *iA 
  2 
  - 
  e 
  l2 
  2 
  = 
  ~ 
  2n 
  ( 
  p 
  - 
  n) 
  {a 
  n 
  e 
  22 
  + 
  a 
  22 
  c 
  n 
  - 
  2a 
  l2 
  c 
  l2 
  

  

  - 
  2n 
  2 
  {a 
  n 
  a 
  22 
  - 
  a 
  12 
  2 
  ) 
  } 
  + 
  inb 
  22 
  {c 
  u 
  - 
  ii 
  2 
  a 
  u 
  ) 
  

  

  + 
  inb 
  22 
  (c 
  u 
  — 
  n 
  2 
  a 
  u 
  ), 
  (92) 
  

  

  use 
  being 
  made 
  of 
  (89). 
  

  

  From 
  (91) 
  with 
  use 
  of 
  (88), 
  

  

  Mod 
  2 
  e 
  V2 
  =(c 
  n 
  —n 
  2 
  a 
  n 
  )(c 
  22 
  —n\ 
  2 
  ); 
  . 
  . 
  . 
  (93) 
  

   and 
  from 
  (92) 
  

  

  tfod 
  2 
  (^ 
  « 
  = 
  Mrf^ 
  {c 
  n 
  c 
  2 
  ,-y-n^o^-O 
  } 
  2 
  

  

  + 
  n% 
  2 
  (c 
  u 
  -ri 
  2 
  a 
  n 
  y. 
  . 
  (94) 
  

  

  If 
  we 
  now, 
  as 
  for 
  (84), 
  carry 
  out 
  the 
  integration 
  with 
  respect 
  

   to 
  p, 
  Mod 
  S 
  P 
  2 
  being 
  constant, 
  we 
  find 
  from 
  (85) 
  

  

  ^Mod 
  2 
  ^^ 
  7r(c 
  22 
  — 
  n 
  2 
  a 
  22 
  ) 
  , 
  

  

  Mod 
  2 
  ^ 
  2 
  2b 
  22 
  \c 
  n 
  c 
  22 
  — 
  c 
  v 
  ?—n\a 
  n 
  a 
  22 
  —a 
  v 
  2 
  ) 
  \ 
  ' 
  

  

  So 
  far 
  we 
  have 
  assumed 
  merely 
  that 
  the 
  compound 
  system 
  

   is 
  in 
  high 
  resonance 
  when 
  p 
  = 
  n 
  ; 
  but 
  more 
  than 
  this 
  is 
  

   required 
  in 
  order 
  to 
  arrive 
  at 
  a 
  simple 
  result. 
  We 
  must 
  

   further 
  assume 
  that 
  the 
  coefficients 
  of 
  interconnexion 
  a 
  m 
  

  

  