﻿Combination 
  with 
  Homogeneous 
  Functions. 
  469 
  

  

  As 
  a 
  simple 
  example 
  let 
  it 
  be 
  required 
  to 
  expand 
  x 
  4 
  in 
  the 
  

   form 
  

  

  ^ 
  = 
  H 
  4 
  + 
  r 
  2 
  H 
  2 
  + 
  r 
  4 
  H 
  , 
  

  

  where 
  the 
  subscripts 
  denote 
  the 
  degrees 
  of 
  the 
  harmonics, 
  

   Operating 
  with 
  A 
  an 
  d 
  determining 
  the 
  numerical 
  coefficients 
  

   on 
  the 
  right 
  by 
  (F), 
  

  

  A.* 
  4 
  - 
  12.i- 
  2 
  = 
  14H 
  2 
  + 
  20r 
  2 
  H 
  ; 
  A 
  2 
  -* 
  4 
  = 
  24= 
  120H 
  , 
  

   whence 
  

   H 
  =l, 
  H 
  2 
  = 
  |(6^-2r 
  2 
  ), 
  H 
  4 
  = 
  ^-3r 
  2 
  (6^ 
  2 
  -2r 
  2 
  )-ir 
  4 
  . 
  

  

  It 
  appears 
  that, 
  in 
  general, 
  each 
  of 
  the 
  H's 
  will 
  be 
  expressed 
  

   in 
  terms 
  of 
  all 
  the 
  ri's 
  of 
  lower 
  subscript. 
  

  

  10. 
  Term 
  by 
  term 
  evaluation 
  of 
  the 
  integral. 
  — 
  Let 
  us 
  now 
  

   use 
  the 
  elementary 
  theory 
  of 
  the 
  potential 
  function 
  to 
  evaluate 
  

   the 
  integral 
  

  

  JO 
  

  

  ? 
  pf(r) 
  • 
  HrfV 
  

  

  over 
  a 
  sphere 
  of 
  radius 
  a 
  with 
  centre 
  at 
  the 
  origin. 
  This 
  is 
  

   the 
  same 
  as 
  finding 
  the 
  potential 
  at 
  a 
  point 
  0' 
  due 
  to 
  a 
  

   volume-density 
  of 
  f(r) 
  . 
  H„ 
  throughout 
  the 
  sphere. 
  Suppose 
  

   first 
  that 
  0' 
  is 
  outside 
  the 
  sphere. 
  Following 
  Maxwell's 
  

   notation*, 
  we 
  may 
  write 
  H 
  u 
  = 
  r 
  n 
  Y 
  n 
  , 
  where 
  Y 
  B 
  is 
  a 
  surface 
  

   harmonic. 
  The 
  potential 
  at 
  an 
  external 
  point 
  due 
  to 
  a 
  

   surface-distribution 
  Y 
  n 
  over 
  a 
  sphere 
  of: 
  radius 
  a 
  with 
  

  

  4 
  7ra 
  »+2Y 
  

  

  centre 
  at 
  the 
  orio-in 
  is 
  known 
  to 
  be 
  y^ 
  .,, 
  ".. 
  , 
  that 
  is 
  

  

  4 
  7ra 
  n+2H 
  n 
  # 
  (2n 
  + 
  l)r 
  n+1 
  

  

  (9 
  a. 
  i) 
  2n+n 
  am 
  J? 
  since 
  H 
  n 
  = 
  Y 
  n 
  a 
  n 
  at 
  the 
  surface, 
  a 
  distri- 
  

   bution 
  f(a)K 
  n 
  over 
  the 
  surface 
  will 
  give 
  an 
  external 
  potential 
  

  

  4/ra 
  2,l+2 
  /'(a)H; 
  l 
  

  

  -7-7 
  ;, 
  „ 
  ." 
  -. 
  For 
  the 
  external 
  potential 
  due 
  to 
  the 
  solid 
  

  

  {'2n+ 
  l)r 
  Zn 
  ^ 
  1 
  x 
  

  

  sphere 
  we 
  therefore 
  have 
  

   Formula 
  (H) 
  

  

  J{,-).K„<IY= 
  ( 
  J^, 
  I+ 
  ^W)<0-, 
  ■ 
  (H) 
  

  

  the 
  accents 
  being 
  dropped 
  after 
  the 
  integration, 
  i. 
  e. 
  r 
  is 
  put 
  

   for 
  T(OO'). 
  We 
  can 
  now 
  evaluate 
  the 
  integral 
  propounded 
  

  

  * 
  Elect, 
  and 
  Mag. 
  3rd 
  Ed. 
  vol. 
  i. 
  Art. 
  131 
  a. 
  

  

  jyj> 
  

  

  