﻿Combination 
  with 
  Homogeneous 
  Functions. 
  467 
  

  

  since 
  the 
  new 
  integrand 
  does 
  not 
  contain 
  x. 
  The 
  area 
  is 
  

   therefore 
  given 
  by 
  integration 
  over 
  a 
  semicircle 
  of 
  radius 
  8 
  

   standing 
  on 
  the 
  x 
  axis, 
  by 
  (D), 
  

  

  ff 
  to 
  dA 
  =i 
  . 
  |j>,[J>-0]= 
  1 
  • 
  f. 
  16= 
  ™. 
  

  

  It 
  may 
  fairly 
  be 
  said, 
  however, 
  that 
  we 
  are 
  here 
  intruding 
  

   on 
  ground 
  properly 
  belonging 
  to 
  ordinary 
  rectangular 
  coor- 
  

   dinates. 
  Yet 
  the 
  method 
  may 
  with 
  equal 
  ease 
  be 
  applied 
  to 
  

   problems 
  of 
  much 
  greater 
  complexity; 
  and 
  has 
  some 
  theoretic 
  

   interest 
  in 
  that 
  we 
  use 
  a 
  process 
  of 
  differentiation 
  to 
  arrive 
  

   at 
  the 
  definite 
  integral. 
  

  

  8. 
  A 
  more 
  general 
  problem. 
  — 
  We 
  may 
  now 
  travel 
  a 
  little 
  

   further 
  afield 
  and 
  bring 
  the 
  foregoing 
  integrals 
  in 
  touch 
  with 
  

   the 
  theory 
  of 
  the 
  potential 
  function. 
  As 
  a 
  first 
  step 
  in 
  this 
  

   direction 
  let 
  it 
  be 
  required 
  to 
  extend 
  formula 
  (A) 
  so 
  as 
  to 
  

   evaluate 
  

  

  ffijftr) 
  .F(p)dY, 
  

  

  (7) 
  

  

  where 
  r' 
  is 
  the 
  distance 
  from 
  the 
  variable 
  point 
  p 
  to 
  some 
  

   fixed 
  point 
  other 
  than 
  the 
  origin. 
  This 
  is 
  the 
  same 
  as 
  finding 
  

   the 
  potential, 
  at 
  the 
  fixed 
  point, 
  due 
  to 
  the 
  attraction 
  of 
  the 
  

   sphere 
  if 
  its 
  density 
  at 
  p 
  is/(r) 
  .F(p). 
  The 
  homogeneous 
  

   function 
  F(yo) 
  is, 
  however, 
  assumed 
  at 
  present 
  to 
  be 
  a 
  

   polynomial. 
  

  

  9. 
  Theorem 
  on 
  the 
  expansion 
  of 
  a 
  polynomial 
  in 
  terms 
  of 
  

   harmonics. 
  — 
  The 
  solution 
  of 
  the 
  above 
  problem 
  depends 
  on 
  

   the 
  following 
  theorem: 
  — 
  

  

  Any 
  homogeneous 
  polynomial 
  Y 
  m 
  (p) 
  differs 
  from 
  an 
  harmonic 
  

   function 
  by 
  a 
  sum 
  of 
  terms, 
  each 
  of 
  which 
  consists 
  of 
  an 
  even 
  

   power 
  of 
  the 
  radius 
  vector 
  multiplied 
  by 
  an 
  harmonic 
  function 
  of 
  

   lower 
  degree. 
  

  

  In 
  symbols 
  this 
  theorem 
  may 
  be 
  stated 
  as 
  

  

  Formula 
  (E) 
  

  

  F 
  m 
  =H„ 
  1 
  + 
  r 
  2 
  H„ 
  1 
  . 
  2 
  +^H 
  M 
  _ 
  4 
  +... 
  + 
  »-H 
  . 
  . 
  (E) 
  

  

  if 
  m 
  is 
  even; 
  while 
  if 
  m 
  is 
  odd 
  the 
  last 
  term 
  is 
  r"*" 
  1 
  ^. 
  Here 
  

   the 
  H's 
  are 
  harmonic 
  polynomials 
  of 
  degrees 
  indicated 
  by 
  

   their 
  subscripts. 
  

  

  To 
  prove 
  this 
  result 
  w 
  r 
  e 
  have 
  first 
  to 
  find 
  the 
  effect 
  of 
  V 
  

   on 
  any 
  term 
  of 
  the 
  form 
  r 
  k 
  t, 
  where 
  t 
  is 
  any 
  homogeneous 
  

   scalar 
  function 
  of 
  degree 
  n 
  in 
  p. 
  By 
  direct 
  expansion, 
  (re- 
  

   membering 
  V?' 
  = 
  m), 
  

  

  S7(rH)=kr 
  k 
  - 
  1 
  ut 
  + 
  r 
  A 
  Vt=kr 
  k 
  - 
  2 
  pt 
  + 
  r*S7t, 
  . 
  . 
  (8) 
  

  

  