﻿between 
  Cathode 
  and 
  Rontgen 
  Rays. 
  175 
  

  

  ewy 
  3ew" 
  2 
  yzt 
  

  

  a= 
  — 
  

  

  As 
  we 
  are 
  neglecting 
  iv 
  2 
  we 
  may 
  leave 
  out 
  the 
  second 
  terms 
  

   in 
  these 
  equations. 
  These 
  values 
  hold 
  from 
  t 
  — 
  to 
  

   t=(r-a)/Y. 
  When 
  t>(r 
  + 
  a)/V, 
  

  

  We 
  must 
  now 
  allow 
  for 
  the 
  absence 
  of 
  magnetic 
  force 
  

   inside 
  the 
  sphere 
  of 
  radius 
  a 
  ; 
  the 
  easiest 
  way 
  to 
  do 
  this 
  is 
  to 
  

   suppose 
  that 
  the 
  expressions 
  (1) 
  hold 
  right 
  up 
  to 
  the 
  centre 
  of 
  

   this 
  sphere, 
  and 
  superpose 
  on 
  the 
  distribution 
  represented 
  by 
  

   (1) 
  a 
  distribution 
  inside 
  the 
  sphere 
  given 
  by 
  

  

  ewy 
  _ 
  ewx 
  /ox 
  

  

  a 
  =7ib 
  ^-^r, 
  .... 
  (3) 
  

  

  where 
  r* 
  = 
  x 
  2 
  + 
  y 
  2 
  + 
  z 
  2 
  ; 
  

  

  while 
  outside 
  the 
  sphere 
  we 
  have 
  for 
  this 
  distribution 
  

  

  If 
  we 
  superpose 
  this 
  distribution 
  we 
  may 
  suppose 
  that 
  at 
  any 
  

   time 
  

  

  where 
  a 
  x 
  , 
  ft 
  are 
  the 
  values 
  given 
  by 
  equations 
  (2) 
  which 
  

   may 
  be 
  now 
  supposed 
  to 
  hold 
  from 
  t 
  = 
  Q 
  to 
  t 
  = 
  r/Y, 
  while, 
  

   when 
  t 
  > 
  r/V, 
  » 
  l9 
  ft 
  both 
  vanish 
  ; 
  u 
  2 
  , 
  ft 
  are 
  the 
  magnetic 
  

   forces 
  arising 
  from 
  the 
  disturbance 
  given 
  initially 
  by 
  (3). 
  

   This 
  disturbance 
  will 
  begin 
  to 
  be 
  felt 
  at 
  a 
  point 
  P 
  at 
  a 
  time 
  

   (OP 
  — 
  a)/Y, 
  and 
  will 
  cease 
  after 
  a 
  time 
  (OP 
  + 
  a)/V; 
  is 
  the 
  

   centre 
  of 
  the 
  charged 
  sphere. 
  Thus 
  the 
  thickness 
  of 
  the 
  

   pulse 
  due 
  to 
  this 
  distribution 
  is 
  equal 
  to 
  the 
  diameter 
  of 
  the 
  

   sphere. 
  

  

  We 
  can 
  easily 
  show 
  that 
  

  

  taken 
  over 
  the 
  part 
  of 
  a 
  sphere 
  whose 
  centre 
  is 
  at 
  P 
  and 
  

   radius 
  is 
  Yt, 
  which 
  is 
  within 
  the 
  sphere 
  whose 
  radius 
  is 
  a, 
  is 
  

   equal 
  to 
  

  

  27rwe 
  . 
  YH 
  2 
  f 
  (0P 
  2 
  -q3-V 
  2 
  ^ 
  _#_ 
  

   OP 
  2 
  I 
  2Yt 
  . 
  a 
  ) 
  OP' 
  

  

  where 
  x 
  is 
  the 
  x 
  coordinate 
  of 
  P. 
  

  

  02 
  

  

  