﻿betiveen 
  Cathode 
  and 
  Rontgen 
  Rays. 
  Ill 
  

  

  the 
  line 
  integral 
  is 
  equal 
  to 
  the 
  change 
  in 
  the 
  displacement 
  ; 
  

   but 
  when 
  we 
  neglect 
  io 
  2 
  / 
  V 
  2 
  , 
  the 
  distribution 
  of 
  the 
  displace- 
  

   ment 
  is 
  the 
  same 
  when 
  the 
  sphere 
  is 
  moving 
  as 
  when 
  in 
  the 
  

   steady 
  state 
  at 
  rest 
  : 
  thus 
  the 
  time 
  integral 
  must 
  vanish. 
  

  

  Let 
  us 
  now 
  consider 
  the 
  case 
  when 
  the 
  velocity 
  of 
  the 
  

   particles 
  is 
  nearly 
  equal 
  to 
  that 
  of 
  light. 
  In 
  the 
  limiting 
  case 
  

   when 
  w 
  — 
  Y 
  we 
  see, 
  from 
  the 
  expressions 
  given 
  for 
  a, 
  ft, 
  that 
  

   they 
  vanish 
  unless 
  2=0, 
  when 
  they 
  become 
  infinite 
  ; 
  in 
  this 
  

   case 
  the 
  original 
  magnetic 
  field 
  is 
  confined 
  to 
  a 
  plane 
  through 
  

   the 
  centre 
  of 
  the 
  sphere 
  at 
  right 
  angles 
  to 
  its 
  direction 
  of 
  

   motion. 
  When 
  w 
  is 
  nearly 
  but 
  not 
  quite 
  equal 
  to 
  V, 
  the 
  dis- 
  

   turbance 
  is 
  practically 
  confined 
  between 
  the 
  two 
  cones 
  whose 
  

  

  IT 
  IT 
  

  

  semi-vertical 
  angles 
  are 
  ~ 
  — 
  S 
  and 
  ~- 
  + 
  3, 
  where 
  $ 
  is 
  a 
  small 
  

  

  angle. 
  To 
  simplify 
  the 
  analysis 
  and 
  yet 
  retain 
  the 
  essential 
  

   physical 
  features 
  of 
  the 
  case, 
  we 
  shall 
  suppose 
  that 
  the 
  initial 
  

   disturbance, 
  instead 
  of 
  being 
  confined 
  between 
  these 
  two 
  

   cones, 
  is 
  confined 
  between 
  the 
  planes 
  z~- 
  + 
  d 
  and 
  2= 
  — 
  d, 
  where 
  

   d 
  is 
  a 
  small 
  quantity; 
  and 
  that 
  both 
  the 
  magnetic 
  force 
  and 
  

   the 
  electric 
  intensity 
  are 
  parallel 
  to 
  the 
  planes, 
  the 
  lines 
  of 
  

   electric 
  intensity 
  being 
  radial 
  at 
  right 
  angles 
  to 
  the 
  axis 
  of 
  z, 
  

   and 
  the 
  lines 
  of 
  magnetic 
  force 
  circles 
  with 
  their 
  centres 
  on 
  

   the 
  axis 
  of 
  z. 
  Let 
  E 
  be 
  the 
  electric 
  intensity 
  at 
  a 
  point 
  distant 
  

   p 
  from 
  this 
  axis 
  ; 
  then 
  the 
  total 
  normal 
  induction 
  over 
  the 
  

   surface 
  of 
  a 
  cylinder 
  passing 
  through 
  this 
  point 
  and 
  with 
  

   the 
  axis 
  of 
  z 
  for 
  its 
  axis 
  is 
  equal 
  to 
  

  

  E 
  x 
  2irp 
  x 
  2d, 
  

  

  this 
  must 
  equal 
  kire 
  ; 
  hence 
  

  

  dp 
  

  

  Hence 
  if 
  a, 
  /3 
  are 
  the 
  components 
  of 
  the 
  magnetic 
  force 
  just 
  

   after 
  the 
  particle 
  is 
  stopped, 
  

  

  0= 
  

  

  Yey 
  

  

  ' 
  ay' 
  

  

  Yex 
  

  

  CL-- 
  ^-5-, 
  

  

  dp 
  

  

  2 
  1 
  

  

  doc 
  _ 
  v 
  ^_ 
  d[3 
  _ 
  _ 
  v 
  d/3 
  

   dt 
  ~ 
  dz' 
  dt 
  dz 
  

  

  Both 
  dcc/dt 
  and 
  dft/dt 
  are 
  zero 
  except 
  when 
  z— 
  ±d, 
  when 
  

   they 
  are 
  infinite. 
  

  

  These 
  equations 
  give 
  the 
  initial 
  state 
  of 
  the 
  field 
  outside 
  the 
  

   charged 
  particle 
  ; 
  inside 
  this 
  particle, 
  which 
  we 
  shall 
  take 
  to 
  

  

  