﻿764 
  Prof. 
  0. 
  W. 
  Richardson 
  on 
  the 
  Specific 
  

  

  to 
  the 
  axis 
  of 
  the 
  cylinder 
  ; 
  let 
  us 
  calculate 
  the 
  sideways 
  

   displacement 
  x 
  of 
  the 
  moving 
  ion 
  due 
  to 
  the 
  action 
  of 
  the 
  

   magnetic 
  force, 
  whose 
  direction 
  is 
  taken 
  to 
  be 
  that 
  of 
  the 
  

   y 
  axis. 
  

  

  The 
  radial 
  electric 
  force 
  at 
  a 
  distance 
  v 
  from 
  the 
  axis 
  is 
  

  

  Z=Y/slog 
  b/a. 
  

  

  The 
  equations 
  of 
  motion 
  of 
  the 
  ions 
  are 
  

  

  d 
  2 
  z 
  r/ 
  jj 
  dx 
  

  

  and 
  d 
  2 
  x 
  „ 
  dz 
  

  

  m 
  M= 
  Re 
  dt' 
  

  

  dx 
  

   Integrating 
  the 
  last 
  subject 
  to 
  the 
  condition 
  -77 
  =0, 
  when 
  

  

  z 
  = 
  a 
  we 
  have 
  

  

  dx 
  He 
  s 
  s 
  

  

  - 
  = 
  — 
  (z 
  — 
  a), 
  

   dt 
  m 
  K 
  J 
  ' 
  

  

  so 
  that 
  ^ 
  Ye 
  HV 
  _ 
  

  

  m 
  di? 
  ~~ 
  Zlogb/a 
  m 
  {Z 
  a) 
  ' 
  

  

  Integrating 
  this 
  subject 
  to 
  the 
  condition 
  that 
  when 
  Z 
  = 
  a, 
  

  

  — 
  = 
  iv. 
  we 
  obtain 
  

   dt 
  

  

  (dz\ 
  2 
  _ 
  9 
  2V* 
  logs/a 
  HV 
  __ 
  

   \dt) 
  ~ 
  W 
  + 
  mlogrb/a 
  m? 
  [Z 
  a 
  >' 
  

  

  Since 
  „ 
  , 
  x 
  <fo 
  <fe 
  & 
  

  

  Hi(f.a) 
  B 
  » 
  3 
  =» 
  5 
  5 
  

  

  we 
  have 
  finally, 
  the 
  sideways 
  displacement 
  

  

  He 
  f 
  b 
  (z-a)dz 
  

  

  x 
  1 
  = 
  — 
  

  

  ™ 
  Ja 
  

  

  L 
  im 
  log 
  6/a 
  m 
  2 
  'J 
  

  

  The 
  corresponding 
  expression 
  for 
  planes, 
  distance 
  b 
  — 
  a 
  

   apart, 
  is 
  

  

  1 
  1_ 
  * 
  «l 
  Jo 
  / 
  2 
  1 
  

  

  WM 
  71 
  s 
  — 
  —7T-Z 
  

  

  2Yez 
  _HV 
  A*' 
  

  

  (b 
  — 
  a) 
  ?n 
  2 
  

  

  To 
  compare 
  with 
  the 
  experimental 
  results 
  these 
  integrals 
  

   have 
  been 
  determined 
  graphically 
  and 
  their 
  values 
  compared 
  

  

  