﻿

  Fig. 
  1. 
  

  

  

  if 
  

  

  b 
  if 
  b 
  

  

  if 
  

  

  b 
  

  

  if 
  b 
  if 
  

  

  b 
  

  

  if 
  

  

  b 
  if 
  b 
  

  

  if 
  

  

  b 
  

  

  if 
  b 
  if 
  

  

  b 
  

  

  356 
  Mr. 
  W. 
  Sutherland 
  on 
  

  

  The 
  results 
  of 
  Barus 
  will 
  be 
  discussed 
  soon 
  in 
  connexion 
  

   with 
  the 
  following 
  attempt 
  to 
  carry 
  the 
  theoretical 
  equations 
  

   of 
  Thomson 
  and 
  Langevin 
  to 
  a 
  higher 
  degree 
  of 
  approxima- 
  

   tion. 
  If 
  the 
  ions 
  were 
  always 
  distributed 
  symmetrically, 
  

   each 
  would 
  be 
  in 
  statical 
  equilibrium 
  as 
  regards 
  the 
  electric 
  

   forces, 
  and 
  the 
  beginnings 
  of 
  recombination 
  would 
  never 
  

   arise. 
  Recombination 
  depends 
  chiefly 
  upon 
  the 
  rate 
  at 
  which 
  

   unsymmetrical 
  positions 
  are 
  formed 
  during 
  the 
  motion 
  of 
  the 
  

   ions. 
  Let 
  ns 
  starts 
  then, 
  with 
  a 
  convenient 
  specification 
  of 
  

   the 
  symmetrical 
  positions. 
  Suppose 
  the 
  space 
  divided 
  into 
  

   equal 
  cubes 
  of 
  edge 
  R 
  with 
  an 
  ion 
  placed 
  at 
  each 
  corner 
  

   which 
  is 
  common 
  to 
  8 
  cubes, 
  the 
  ions 
  

   occurring 
  alternately 
  negative 
  and 
  

   positive 
  along 
  the 
  straight 
  lines 
  formed 
  

   by 
  the 
  edges 
  of 
  the 
  cubes 
  according 
  to 
  

   the 
  specimen 
  plan 
  of 
  fig. 
  1. 
  Like 
  ions 
  

   are 
  arranged 
  along 
  diagonals. 
  If 
  any 
  

   ion 
  is 
  displaced 
  from 
  a 
  diagonal, 
  it 
  will 
  

   be 
  attracted, 
  most 
  strongly 
  towards 
  

   the 
  nearest 
  diagonal. 
  Let 
  us 
  then 
  

   replace 
  the 
  electrons 
  by 
  lines 
  of 
  elec- 
  if 
  b 
  if 
  b 
  if 
  

   tricity 
  of 
  density 
  6/2^/^11 
  along 
  diagonals 
  

   running 
  downwards 
  from 
  left 
  to 
  right. 
  

  

  Then 
  in 
  fig. 
  2 
  let 
  us 
  make 
  a 
  section 
  of 
  these 
  parallel 
  lines 
  by 
  

   a 
  plane 
  at 
  right 
  angles 
  to 
  them, 
  the 
  intersections 
  of 
  the 
  lines 
  

   and 
  plane 
  being 
  marked 
  # 
  for 
  the 
  

   positive 
  lines 
  and 
  b 
  for 
  the 
  negative. 
  ^ig- 
  2. 
  

  

  Again 
  # 
  ranges 
  in 
  diagonal 
  lines, 
  if 
  >* 
  

  

  and 
  so 
  does 
  b. 
  These 
  diagonals 
  are 
  ^ 
  ^ 
  if 
  j* 
  

  

  at 
  distance 
  R/3^''^ 
  apart, 
  while 
  the 
  :Jf 
  ^ 
  ^ 
  ^ 
  

  

  distance 
  between 
  # 
  and 
  its 
  neighbour 
  b 
  b 
  ^ 
  it 
  

  

  t 
  is 
  R(3/2)i/2. 
  Spread 
  the 
  lines 
  of» 
  ^^^ 
  ^ 
  

   density 
  e/2'^^^ 
  R 
  so 
  that 
  they 
  become 
  b 
  ,^ 
  Jf 
  

  

  planes 
  of 
  surface 
  density 
  ^/3^/^R^, 
  b 
  ^ 
  

  

  these 
  planes 
  being 
  alternately 
  jt 
  and 
  

  

  \}. 
  In 
  this 
  way 
  we 
  have 
  substituted 
  for 
  the 
  original 
  point 
  

   distribution 
  an 
  equiyalent 
  laminar 
  one. 
  We 
  can 
  now 
  treat 
  

   the 
  problem 
  of 
  recombination 
  of 
  ions 
  as 
  one 
  of 
  leakage 
  in 
  

   our 
  laminar 
  distribution. 
  Writing 
  the 
  intensity 
  of 
  the 
  

   electric 
  force 
  between 
  two 
  laminse 
  as 
  4:7re/S^^^W, 
  we 
  consider 
  

   the 
  rate 
  of 
  leak 
  proportional 
  to 
  this 
  force 
  and 
  to 
  the 
  mobility 
  

   Ui 
  or 
  U2 
  ol 
  the 
  ions, 
  and 
  to 
  Ni. 
  But 
  NiR^ 
  = 
  1, 
  so 
  with 
  A 
  a 
  

   parameter 
  we 
  have 
  finally 
  

  

  dl^,/dt 
  = 
  -A'N,'lK 
  ...... 
  (17) 
  

  

  This 
  factor 
  of 
  proportionality 
  A 
  is 
  proportional 
  to 
  Ui 
  + 
  U2y 
  

  

  