﻿104 
  Lord 
  Rayleigh 
  on 
  some 
  General 
  Theorems 
  

  

  and 
  the 
  maximum 
  work 
  abstractable 
  will 
  be 
  

  

  ^ 
  R 
  " 
  ° 
  r 
  ~ 
  S\ 
  Ai- 
  (37) 
  

  

  The 
  symbolic 
  expression 
  for 
  ^ 
  becomes 
  at 
  the 
  same 
  time 
  

   from 
  (22) 
  

  

  ♦l^l-i 
  + 
  itcotfln, 
  .... 
  (38) 
  

  

  so 
  that 
  the 
  part 
  of 
  ^ 
  in 
  the 
  same 
  phase 
  as 
  when 
  \I/\ 
  = 
  is 
  

   half 
  as 
  great 
  as 
  before. 
  

  

  So 
  far 
  as 
  the 
  results 
  embodied 
  in 
  (37), 
  (38), 
  and 
  (39) 
  are 
  

   concerned, 
  it 
  is 
  a 
  matter 
  of 
  indifference 
  whether 
  the 
  pre- 
  

   scribed 
  ^i 
  = 
  l 
  when 
  ^ 
  = 
  is 
  due 
  to 
  W 
  2 
  only^ 
  or 
  to 
  "^ 
  acting 
  

   in 
  conjunction 
  with 
  forces 
  M^, 
  &c., 
  corresponding 
  to 
  further 
  

   degrees 
  of 
  freedom. 
  But 
  for 
  the 
  present 
  we 
  will 
  suppose 
  

   that 
  there 
  are 
  only 
  two 
  degrees 
  of 
  freedom. 
  

  

  The 
  maximum 
  work 
  that 
  can 
  be 
  abstracted 
  by 
  M^, 
  when 
  tp 
  2 
  

   is 
  given, 
  may 
  be 
  expressed 
  as 
  

  

  ^Mod 
  2 
  ^ 
  L 
  

  

  8A 
  11 
  sina 
  11 
  ' 
  v 
  ' 
  

  

  where 
  ^ 
  is 
  due 
  to 
  M* 
  2 
  acting 
  alone. 
  By 
  (33) 
  

   Mod"^==A 
  u 
  a 
  Mod 
  s 
  *„ 
  

  

  and 
  by 
  (20) 
  with 
  & 
  n 
  =0, 
  6 
  12 
  =0 
  3 
  

  

  — 
  p~ 
  l 
  A 
  n 
  sin 
  a 
  n 
  = 
  b 
  22 
  A 
  21 
  2 
  , 
  

   so 
  that 
  the 
  maximum 
  work 
  obtainable 
  is 
  simply 
  

  

  M 
  -s? 
  « 
  

  

  That 
  it 
  is 
  independent 
  not 
  onry 
  of 
  a 
  n 
  , 
  e 
  n 
  , 
  a 
  22 
  , 
  c 
  22 
  , 
  but 
  also 
  of 
  

   the 
  coefficients 
  of 
  mutual 
  influence 
  a 
  12 
  , 
  c 
  12 
  , 
  is 
  very 
  remarkable. 
  

   To 
  revert 
  to 
  the 
  electrical 
  example, 
  the 
  work 
  abstractable 
  in 
  

   the 
  first 
  circuit 
  (devoid 
  of 
  resistance) 
  when 
  a 
  given 
  electro- 
  

   motive 
  force 
  acts 
  in 
  the 
  second, 
  is 
  independent 
  of 
  the 
  value 
  

   of 
  the 
  coefficient 
  of 
  mutual 
  induction. 
  If 
  indeed 
  this 
  

   coefficient 
  be 
  very 
  small, 
  the 
  supposition 
  of 
  zero 
  resistance 
  

   becomes 
  more 
  and 
  more 
  unpractical 
  on 
  account 
  of 
  the 
  large 
  

   currents 
  which 
  must 
  then 
  be 
  supposed 
  to 
  flow. 
  But 
  the 
  theo- 
  

   retical 
  result 
  remains 
  true, 
  when 
  b 
  u 
  is 
  diminished 
  without 
  

   limit. 
  In 
  view 
  of 
  its 
  independence 
  of 
  so 
  many 
  circumstances 
  

   that 
  might 
  at 
  first 
  be 
  supposed 
  material, 
  it 
  may 
  now 
  not 
  be 
  

   surprising 
  to 
  note 
  that 
  (40) 
  coincides 
  with 
  (31), 
  that 
  is 
  that 
  

   the 
  work 
  obtainable 
  in 
  the 
  second 
  circuit 
  is 
  the 
  same 
  as 
  might 
  

   have 
  been 
  obtained 
  in 
  the 
  first 
  where 
  the 
  given 
  force 
  itself 
  

   acts. 
  

  

  