﻿between 
  Cathode 
  and 
  Rontgen 
  Rays. 
  179 
  

  

  In 
  finding 
  the 
  mean 
  value 
  of 
  ft 
  over 
  the 
  sphere 
  we 
  must 
  

   double 
  this 
  value, 
  for 
  to 
  each 
  value 
  of 
  <ft 
  and 
  z 
  there 
  correspond 
  

   two 
  elements 
  of 
  the 
  surface 
  of 
  the 
  sphere 
  which 
  contribute 
  

   equally 
  to 
  the 
  integral 
  ; 
  hence 
  

  

  aY 
  rr*™' 
  1 
  ^ 
  

  

  a\ 
  rr 
  sm 
  

  

  J/3dS 
  = 
  4e-^- 
  cos 
  j> 
  d(f> 
  dz 
  

  

  J 
  Jo 
  7>\/sin 
  2 
  S 
  — 
  sin 
  

  

  6>/sin 
  2 
  3— 
  sin 
  2 
  <ft 
  

  

  Now 
  the 
  limits 
  of 
  z 
  depend 
  upon 
  whether 
  the 
  sphere 
  does 
  

   not 
  or 
  does 
  cut 
  right 
  through 
  the 
  slab 
  between 
  the 
  two 
  parallel 
  

   planes 
  ; 
  in 
  the 
  former 
  case 
  Vt 
  is 
  less 
  than 
  c 
  + 
  d, 
  and 
  the 
  limits 
  

   of 
  z 
  are 
  c~Yt 
  and 
  d 
  ; 
  then 
  

  

  J 
  /3dS=27re-^(d-c 
  + 
  Yt); 
  

  

  in 
  the 
  latter 
  case 
  Yt 
  is 
  greater 
  than 
  c 
  + 
  d, 
  and 
  the 
  limits 
  of 
  z 
  

   are 
  — 
  d 
  and 
  + 
  d 
  ; 
  hence 
  in 
  this 
  case 
  

  

  §0dS=27re^2d. 
  

  

  Hence 
  e* 
  l9 
  the 
  mean 
  value 
  of 
  the 
  initial 
  value 
  of 
  ft 
  over 
  the 
  

   surface 
  of 
  this 
  sphere, 
  is 
  

  

  1 
  e.Y 
  \d--c 
  + 
  Yt\ 
  

  

  2~bd 
  Yt 
  

  

  in 
  the 
  first 
  case, 
  and 
  

  

  V 
  

   e 
  b.Yt 
  

  

  in 
  the 
  second 
  ; 
  hence 
  

  

  d 
  . 
  . 
  leY 
  n 
  

   dt 
  K 
  v 
  2 
  db 
  

  

  according 
  as 
  V* 
  < 
  or 
  > 
  c 
  + 
  d. 
  

  

  This 
  value 
  of 
  -^ 
  (tco^ 
  is 
  the 
  same 
  whether 
  the 
  point 
  P 
  is 
  in 
  

  

  front 
  or 
  behind 
  the 
  plane. 
  

  

  We 
  now 
  proceed 
  to 
  find 
  the 
  value 
  of 
  

  

  J 
  dt 
  JJ 
  dz 
  coso 
  

  

  Now 
  dft/dz 
  is 
  zero 
  except 
  at 
  the 
  surface 
  of 
  the 
  plane 
  ; 
  henco 
  

  

  