﻿Interaction 
  of 
  Dynamical 
  Systems. 
  

   the 
  resulting 
  values 
  are 
  given 
  by 
  

  

  119 
  

  

  Si 
  = 
  S 
  f" 
  C 
  ll*l 
  + 
  <?12*2 
  + 
  • 
  

  

  where 
  

  

  

  1 
  

  

  J- 
  (24) 
  

  

  J 
  

  

  Here 
  ffj 
  <x 
  2 
  ... 
  refer 
  to 
  the 
  surfaces 
  of 
  barriers, 
  (Imn) 
  are 
  

   direction-cosines 
  of 
  the 
  normal 
  to 
  a 
  barrier, 
  and 
  dp 
  an 
  element 
  

   of 
  this 
  normal. 
  The 
  second 
  suffix 
  in 
  c 
  refers 
  to 
  the 
  barrier, 
  

   the 
  first 
  to 
  the 
  velocities 
  wi... 
  6 
  in 
  order. 
  If 
  we 
  write 
  the 
  

   bilinear 
  function 
  I 
  = 
  % 
  c 
  a 
  bU 
  a 
  K 
  b 
  , 
  the 
  momenta 
  are 
  given 
  by 
  

  

  a,b 
  

  

  (25) 
  

  

  Also 
  if 
  we 
  set 
  down 
  the 
  value 
  of 
  I 
  having 
  regard 
  to 
  the 
  

   meanings 
  of 
  c 
  in 
  (24), 
  we 
  find 
  

  

  I=.X^(lV 
  + 
  mV 
  + 
  nW-^) 
  Kh 
  da 
  b 
  , 
  . 
  . 
  (26) 
  

  

  where 
  TJ—u 
  + 
  qz— 
  ry.. 
  , 
  i.e. 
  (UVW) 
  represents 
  the 
  velocity 
  

   of 
  a 
  point 
  of 
  a 
  barrier 
  supposed 
  to 
  be 
  in 
  rigid 
  connexion 
  with 
  

   the 
  solid. 
  

  

  Now 
  taking 
  k 
  to 
  correspond 
  to 
  ot 
  of 
  the 
  general 
  method, 
  

   if 
  there 
  were 
  no 
  motion 
  other 
  than 
  cyclic 
  we 
  should 
  have 
  

  

  3K 
  rra» 
  

  

  or 
  yj 
  would 
  mean 
  the 
  flux 
  through 
  the 
  barrier 
  due 
  to 
  the 
  cyclic 
  

   motion. 
  The 
  condition 
  that 
  the 
  composite 
  dynamical 
  scheme 
  

   shall 
  be 
  regular 
  requires 
  a 
  modification 
  to 
  

  

  *>=w> 
  {K 
  - 
  l) 
  

  

  cf. 
  (Hid). 
  

  

  Thus 
  

  

  *>=§(w 
  + 
  w- 
  w 
  - 
  mY 
  - 
  nW 
  X 
  d 
  *» 
  • 
  (27) 
  

  

  or 
  it 
  is 
  the 
  flux 
  relative 
  to 
  the 
  moving 
  barrier, 
  due 
  to 
  the 
  

  

  