﻿Interaction 
  of 
  Dynamical 
  Systems. 
  117 
  

  

  coordinates 
  % 
  absent, 
  the 
  momenta 
  ot 
  constant, 
  and 
  the 
  energy 
  

   given 
  as 
  QC^ 
  + 
  POor), 
  and 
  that 
  the 
  only 
  object 
  of 
  referring 
  

   to 
  a 
  homogeneous 
  expression 
  for 
  energy 
  is 
  to 
  ensure 
  the 
  

   regularity 
  of 
  the 
  composite 
  system 
  ; 
  then 
  the 
  method 
  of 
  § 
  3 
  

   gives 
  that 
  assurance 
  without 
  calculating 
  E(£, 
  %). 
  If 
  the 
  

   method 
  of 
  § 
  6 
  is 
  applied, 
  then 
  T 
  and 
  K 
  in 
  (20) 
  are 
  given 
  

   forms 
  and 
  (15) 
  a 
  presumed 
  original. 
  Q 
  22 
  can 
  be 
  constructed 
  

   at 
  once, 
  but 
  not 
  Q 
  n 
  without 
  a 
  knowledge 
  of 
  the 
  coefficients 
  

   of 
  k 
  s 
  the 
  linear 
  function 
  of 
  <?. 
  These 
  are 
  the 
  coefficients 
  of 
  

   Q 
  12 
  , 
  equal 
  in 
  number 
  to 
  those 
  of 
  I 
  (</,*>■), 
  but 
  they 
  are 
  not 
  

   immediately 
  connected 
  with 
  any 
  information 
  we 
  may 
  possess 
  

   as 
  to 
  the 
  mutual 
  action. 
  In 
  this 
  case 
  therefore 
  the 
  method 
  of 
  

   § 
  6 
  is 
  indirect 
  and 
  laborious 
  in 
  comparison 
  with 
  that 
  of 
  § 
  1 
  

   and 
  § 
  3. 
  

  

  But 
  if 
  (15) 
  is 
  an 
  original 
  form 
  which 
  is 
  to 
  be 
  reduced 
  in 
  

   virtue 
  of 
  the 
  absence 
  of 
  % 
  and 
  constancy 
  of 
  «r, 
  then 
  T 
  and 
  K 
  

   in 
  (20) 
  are 
  derived 
  forms 
  and 
  are 
  the 
  Q 
  and 
  P 
  of 
  (iii). 
  The 
  

   bilinear 
  section 
  is 
  in 
  a 
  different 
  form 
  from 
  that 
  of 
  (iii 
  b) 
  , 
  but 
  

   may 
  be 
  connected 
  with 
  it. 
  For 
  if 
  

  

  VI, 
  

  

  so 
  that 
  

  

  Q12 
  = 
  2 
  <y 
  ms 
  4 
  m 
  X 
  s 
  , 
  and 
  K 
  (*r) 
  = 
  2 
  fi 
  rs 
  tsr 
  r 
  'sr 
  s 
  , 
  

  

  m,s 
  r,s 
  

  

  K 
  s 
  = 
  2i 
  y 
  ms 
  q 
  m 
  and 
  * 
  - 
  - 
  = 
  2,£ 
  r 
  

  

  ^ 
  "=Wrr; 
  > 
  . 
  (22) 
  

   then 
  

  

  Its 
  ^ 
  ,, 
  = 
  Z 
  *T 
  r 
  <lmZPrs<Ym8 
  

   s 
  (J"® 
  's 
  m,r 
  s 
  

  

  and 
  the 
  bilinear 
  forms 
  in 
  (21) 
  and 
  (iii 
  6) 
  are 
  identified 
  by 
  

   writing 
  

  

  C 
  mr 
  = 
  t&rs 
  y 
  m 
  * 
  •• 
  (23) 
  

  

  s 
  

  

  If 
  the 
  reader 
  compares 
  this 
  with 
  the 
  work 
  in 
  Thomson 
  & 
  

   Taitfs 
  ' 
  Natural 
  Philosophy 
  > 
  (p. 
  323, 
  § 
  319 
  F'), 
  he 
  will 
  find 
  

   that 
  the 
  coefficients 
  c 
  correspond 
  to. 
  M, 
  N,. 
  . 
  . 
  ; 
  and 
  the 
  transfer 
  

   to 
  this 
  final 
  form 
  is 
  in 
  fact 
  essential 
  to 
  a 
  compact 
  statement 
  

   of 
  the 
  result. 
  Or, 
  in 
  other 
  words, 
  the 
  problem 
  for 
  the 
  remaining 
  

   coordinates 
  is 
  conveniently 
  restated 
  in 
  the 
  form 
  (iii), 
  after 
  

   finding 
  T 
  and 
  K 
  as 
  in 
  (20), 
  and 
  c 
  mr 
  as 
  in 
  (23). 
  

  

  § 
  8. 
  A 
  brief 
  summary 
  of 
  the 
  position 
  may 
  be 
  given. 
  We 
  

   have 
  just 
  shown 
  that 
  the 
  appearance 
  of 
  E 
  as 
  the 
  sum 
  of 
  

   detached 
  energy-forms 
  of 
  unlike 
  type, 
  viz. 
  : 
  

  

  E 
  = 
  Q(2) 
  + 
  PM 
  (iii 
  a) 
  

  

  is 
  consistent 
  with 
  a 
  mutual 
  action. 
  That 
  action 
  is 
  expressed 
  

  

  