﻿470 
  Operator\/ 
  in 
  Combination 
  with 
  Homogeneous 
  Functions. 
  

  

  in 
  Art. 
  8 
  when 
  the 
  point 
  0' 
  is 
  outside 
  the 
  sphere 
  by 
  applying 
  

   (E) 
  and 
  (H) 
  in 
  succession, 
  viz. 
  

  

  To 
  illustrate, 
  let 
  us 
  find 
  the 
  external 
  potential 
  when 
  the 
  

   density 
  of 
  the 
  solid 
  sphere 
  varies 
  as 
  the 
  fourth 
  power 
  of 
  I 
  he 
  

   distance 
  from 
  a 
  diametral 
  plane. 
  By 
  the 
  expansion 
  of 
  # 
  4 
  

   already 
  obtained 
  

  

  JTJ' 
  

  

  %dV 
  

   r 
  

  

  47TH, 
  

  

  a 
  11 
  4ttH 
  2 
  a 
  9 
  , 
  4ttH 
  

  

  a 
  7 
  

  

  9r 
  2 
  

  

  11 
  ^ 
  5r 
  5 
  9 
  r 
  

  

  'y 
  

  

  where 
  the 
  values 
  of 
  the 
  H's 
  are 
  those 
  calculated 
  in 
  Art. 
  9 
  

   for 
  the 
  expansion 
  of 
  «.t' 
  4 
  , 
  and 
  r 
  is 
  written 
  for 
  T(00 
  / 
  ) 
  on 
  the 
  

   right. 
  

  

  11. 
  The 
  potential 
  inside 
  the 
  sphere. 
  — 
  It 
  is 
  known 
  that 
  a 
  

   surface-distribution 
  Y„ 
  over 
  a 
  sphere 
  of 
  radius 
  a 
  and 
  centre 
  

  

  ^ 
  .,nV 
  

  

  at 
  the 
  origin 
  causes 
  a 
  potential 
  inside 
  the 
  sphere 
  r^ 
  — 
  —z 
  . 
  * 
  , 
  

  

  47rH 
  n 
  . 
  ( 
  Zn 
  + 
  l 
  ) 
  a 
  . 
  . 
  

  

  or 
  rc 
  . 
  — 
  --^-^-'—,. 
  The 
  potential 
  at 
  a 
  point 
  inside 
  the 
  solid 
  

   (2n 
  + 
  l)a 
  n 
  l 
  r 
  r 
  

  

  sphere 
  may 
  be 
  regarded 
  as 
  due 
  to 
  two 
  additive 
  causes, 
  first 
  a 
  

   solid 
  sphere 
  of 
  radius 
  r 
  on 
  whose 
  surface 
  the 
  point 
  lies, 
  

   second 
  a 
  shell 
  of 
  thickness 
  a 
  — 
  r 
  fitting 
  outside 
  the 
  first. 
  The 
  

   first 
  part 
  of 
  the 
  resulting 
  potential 
  is, 
  by 
  (H), 
  

  

  4?rH 
  * 
  f 
  r 
  r 
  2n+2 
  fM 
  d 
  . 
  

  

  and 
  the 
  second 
  part, 
  due 
  to 
  a 
  surface-density 
  f(r)H 
  n 
  dr 
  on 
  

   each 
  infinitesimal 
  shell 
  of 
  radius 
  greater 
  than 
  r, 
  is 
  by 
  the 
  

   formula 
  quoted 
  at 
  the 
  beginning 
  of 
  this 
  article, 
  

  

  r^^\ 
  a 
  rf(r)dr. 
  

  

  (2n 
  + 
  l) 
  J 
  r 
  JK 
  J 
  

  

  Hence 
  the 
  potential 
  inside 
  the 
  solid 
  sphere 
  due 
  to 
  volume- 
  

   distribution 
  /(rH), 
  n 
  is 
  given 
  by 
  

   Formula 
  (A') 
  

  

  jjj>,) 
  H„ 
  rf 
  v 
  = 
  0l.) 
  { 
  M: 
  '' 
  2 
  " 
  +2 
  /^ 
  * 
  + 
  £ 
  *w 
  * 
  } 
  - 
  

  

  • 
  • 
  • 
  (H') 
  

   whence 
  by 
  means 
  of 
  the 
  expansion 
  (E) 
  we 
  can 
  find 
  the 
  value 
  

   of 
  (7) 
  when 
  the 
  point 
  is 
  inside 
  the 
  sphere. 
  

  

  