﻿110 
  Mr. 
  R. 
  Hargr 
  eaves 
  on 
  

  

  is 
  produced 
  outside 
  the 
  zone. 
  The 
  water-vapour 
  enters 
  at 
  

   the 
  centre 
  of 
  the 
  spark 
  ; 
  and 
  if 
  no 
  recombination 
  occurs 
  near 
  

   the 
  cathode, 
  equal 
  quantities 
  of 
  oxygen 
  will 
  be 
  swept 
  towards 
  

   each 
  pole, 
  and 
  the 
  hydrogen 
  being 
  uniformly 
  distributed, 
  

   pure 
  electrolytic 
  gas 
  would 
  be 
  collected 
  at 
  both 
  anode 
  and 
  

   cathode. 
  

  

  But 
  recombination 
  is 
  taking 
  place 
  near 
  the 
  cathode, 
  and 
  

   consequently 
  some 
  of 
  the 
  electrolytic 
  gas 
  is 
  removed 
  there, 
  

   so 
  that 
  a 
  smaller 
  quantity 
  of 
  oxygen 
  is 
  swept 
  out 
  at 
  that 
  pole 
  

   than 
  at 
  the 
  anode. 
  The 
  hydrogen, 
  however, 
  still 
  tend.- 
  to 
  

   become 
  distributed 
  uniformly 
  throughout 
  the 
  apparatus, 
  so 
  

   that 
  it 
  now 
  appears 
  in 
  excess 
  at 
  the 
  cathode. 
  

  

  The 
  experiments 
  described 
  in 
  this 
  paper 
  lead 
  us 
  to 
  the 
  con- 
  

   clusion 
  that 
  when 
  electric 
  sparks 
  pass 
  through 
  a 
  compound 
  

   gas 
  such 
  as 
  water-vapour 
  or 
  carbon 
  dioxide, 
  the 
  separation 
  

   and 
  arrangement 
  of 
  the 
  decomposition 
  products 
  is 
  not 
  an 
  

   electrical 
  phenomenon, 
  but 
  results 
  from 
  gaseous 
  diffusion. 
  The 
  

   hypothesis 
  of 
  electrolysis 
  in 
  liquids 
  is 
  therefore 
  inapplicable. 
  

  

  The 
  University, 
  Manchester. 
  

  

  VI. 
  Interaction 
  of 
  Dynamical 
  Systems. 
  

   By 
  R. 
  Hargeeaves 
  *. 
  

  

  IF 
  there 
  are 
  two 
  groups 
  of 
  coordinates 
  q 
  and 
  % 
  with 
  corre- 
  

   sponding 
  velocities 
  q 
  and 
  %\ 
  the 
  kinetic 
  energy 
  of 
  a 
  

   system 
  embracing 
  both 
  groups 
  generally 
  contains 
  terms 
  

   which 
  are 
  products 
  of 
  q 
  and 
  %, 
  as 
  well 
  as 
  functions 
  quadratic 
  

   in 
  the 
  velocities 
  of 
  the 
  separate 
  groups 
  ; 
  and 
  the 
  product- 
  

   terms 
  give 
  a 
  mutual 
  or 
  composite 
  action 
  of 
  the 
  groups 
  in 
  an 
  

   explicit 
  way. 
  This 
  is 
  also 
  the 
  case 
  when 
  the 
  kinetic 
  energy 
  

   is 
  expressed 
  in 
  terms 
  of 
  the 
  momenta. 
  But 
  if 
  the 
  velocities 
  

   of 
  one 
  group 
  and 
  the 
  momenta 
  of 
  the 
  other 
  are 
  used, 
  it 
  is 
  

   characteristic 
  that 
  the 
  energy 
  shall 
  appear 
  as 
  a 
  sum 
  of 
  

   detached 
  quadratics, 
  and 
  all 
  trace 
  of 
  composite 
  action 
  is 
  lost. 
  

  

  It 
  is 
  proposed 
  to 
  deal 
  with 
  this 
  and 
  other 
  cases 
  where 
  a 
  

   mutual 
  action 
  is 
  concealed. 
  For 
  the 
  case 
  stated 
  a 
  kinetic 
  

   potential 
  of 
  mixed 
  type 
  is 
  introduced, 
  i.e. 
  one 
  involving 
  the 
  

   velocities 
  of 
  one 
  group 
  and 
  the 
  momenta 
  of 
  the 
  other. 
  It 
  

   is 
  clear 
  that 
  when 
  coordinates 
  of 
  the 
  second 
  group 
  are 
  

   absent 
  and 
  the 
  momenta 
  invariable, 
  we 
  have 
  the 
  problem 
  of 
  

   ignored 
  coordinates, 
  which 
  therefore 
  appears 
  as 
  a 
  special 
  case. 
  

  

  § 
  1. 
  Potential 
  energy 
  in 
  the 
  form 
  of 
  a 
  function 
  of 
  co- 
  

   ordinates 
  only 
  may 
  be 
  omitted, 
  as 
  its 
  presence" 
  or 
  absence 
  

   does 
  not 
  affect 
  the 
  questions 
  considered. 
  A 
  kinetic 
  energy 
  

  

  * 
  Communicated 
  by 
  the 
  Author. 
  

  

  