﻿892 
  Prof. 
  0. 
  W. 
  RichardsoD 
  on 
  the 
  Kinetic 
  

  

  If 
  V 
  is 
  the 
  difference 
  of 
  potential, 
  and 
  z 
  the 
  distance, 
  be- 
  

   tween 
  the 
  planes 
  V=Zs 
  / 
  . 
  The 
  equations 
  of 
  motion 
  of 
  the 
  

   ions 
  are 
  

  

  Integrating, 
  subject 
  to 
  the 
  conditions 
  that, 
  when 
  

  

  J 
  A 
  dx 
  du 
  dz 
  

  

  '=°- 
  ,//="* 
  ( 
  7T 
  =r 
  «- 
  ,//="' 
  

  

  so 
  that 
  it", 
  when 
  

  

  dx 
  <1>i 
  dz 
  r/ 
  

  

  / 
  = 
  <>. 
  a?=# 
  , 
  // 
  = 
  // 
  u 
  and 
  2r=0, 
  

  

  //,/. 
  y—y 
  =v 
  t, 
  z 
  = 
  u\J 
  + 
  VA 
  i 
  

  

  s 
  

  

  Sine- 
  in 
  the 
  presenl 
  problem 
  everything 
  is 
  independent 
  of 
  ?/ 
  

   we 
  only 
  need 
  to 
  consider 
  the 
  relation 
  between 
  the 
  z 
  and 
  x 
  

   coordinates 
  of 
  the 
  particle. 
  Eliminating 
  the 
  time, 
  we 
  have 
  

  

  .- 
  = 
  >:/. 
  "(■'■-■-y 
  + 
  ."•» 
  (.-,„) 
  . 
  . 
  . 
  (i) 
  

  

  We 
  may 
  regard 
  this 
  equation 
  in 
  two 
  different 
  ways. 
  If 
  

   we 
  star! 
  with 
  assigned 
  values 
  u 
  Q 
  w 
  o£ 
  the 
  velocity 
  com- 
  

   pom 
  nts 
  then 
  the 
  equation 
  gives 
  us 
  the 
  relation 
  between 
  the 
  

   coordinates 
  x 
  and 
  : 
  of 
  any 
  poini 
  in 
  the 
  subsequent 
  path 
  of 
  

   the 
  particle. 
  On 
  the 
  other 
  iiand.il' 
  we 
  consider 
  two 
  fixed 
  

   points, 
  a? 
  , 
  o 
  in 
  A 
  and 
  x, 
  ~ 
  in 
  ( 
  '. 
  the 
  equation 
  (1) 
  will 
  give 
  

   us 
  the 
  relation 
  which 
  has 
  to 
  hold 
  between 
  w 
  and 
  w 
  , 
  for 
  an 
  

   ion 
  emitted 
  from 
  x 
  , 
  to 
  reach 
  x, 
  z' 
  . 
  This 
  relation 
  may 
  be 
  

   expressed 
  more 
  conveniently 
  by 
  solving 
  the 
  equation 
  as 
  a 
  

   quadratic 
  for 
  i/ 
  , 
  which 
  gives 
  

  

  »o 
  = 
  i"o^(lV( 
  1+ 
  lS?)) 
  

  

  (2) 
  

  

  If 
  u 
  has 
  this 
  value 
  the 
  particle 
  will 
  strike 
  the 
  plate 
  D 
  at 
  

   the 
  point 
  x=x, 
  if 
  u 
  is 
  greater 
  than 
  this 
  the 
  particle 
  will 
  go 
  

   to 
  the 
  right, 
  if 
  u 
  is 
  less 
  it 
  will 
  go 
  the 
  left, 
  of 
  x=x. 
  The 
  

   number 
  of 
  particles 
  which 
  go 
  to 
  the 
  right 
  of 
  x 
  = 
  x 
  will 
  there- 
  

   fore 
  be 
  the 
  number 
  of 
  those 
  which 
  have 
  values 
  of 
  u 
  greater 
  

   than 
  

  

  "'('Vt-a?)) 
  

  

  , 
  x 
  — 
  x 
  

   iu'o 
  

  

  