﻿Combination 
  ivith 
  Homogeneous 
  Functions. 
  463 
  

  

  In 
  fact, 
  ^ 
  since 
  A 
  is 
  a 
  scalar 
  operator, 
  the 
  formula 
  can 
  be 
  

   applied 
  in 
  turn 
  to 
  the 
  scalar 
  components 
  of 
  a 
  vector 
  F(/o), 
  

   hence 
  is 
  true 
  for 
  the 
  vector. 
  That 
  is, 
  F(p) 
  in 
  formula 
  (A) 
  

   may 
  be 
  either 
  vector, 
  scalar, 
  or 
  quaternion, 
  /(r) 
  remaining 
  a 
  

   scalar. 
  

  

  4. 
  Extension 
  by 
  geometrical 
  transformation. 
  — 
  Any 
  problem 
  

   which 
  can 
  be 
  worked 
  out 
  by 
  (A) 
  for 
  the 
  sphere 
  maybe 
  made 
  

   to 
  yield 
  an 
  indefinite 
  number 
  of 
  other 
  examples 
  by 
  performing- 
  

   geometrical 
  transformations 
  upon 
  both 
  members 
  of 
  (A). 
  Let 
  

   y(p) 
  be 
  a 
  vector 
  function 
  of 
  (p). 
  If 
  we 
  replace 
  p 
  by 
  y(p) 
  

   and 
  if 
  dy(p)~<f)dp, 
  the 
  Jacobian 
  of 
  the 
  transformation 
  is 
  

   Hamilton's 
  third 
  invariant 
  for 
  the 
  linear 
  vector 
  function 
  <£ 
  

   which 
  in 
  this 
  case 
  I 
  shall 
  call 
  J. 
  The 
  Laplacean 
  operator 
  A, 
  

   which 
  is 
  the 
  same 
  as 
  —V 
  2 
  , 
  is 
  transformed 
  into 
  — 
  ((//^V) 
  2 
  , 
  

   that 
  is 
  to 
  say 
  the 
  operator 
  V 
  is 
  transformed 
  by 
  the 
  reciprocal 
  

   conjugate 
  of 
  the 
  operator 
  <£. 
  And 
  r 
  is 
  transformed 
  into 
  

   '-tyM? 
  the 
  tensor 
  of 
  the 
  transformed 
  point-vector. 
  Formula 
  

   (A)^ 
  now 
  becomes, 
  supposing 
  that 
  we 
  began 
  with 
  a 
  sphere 
  of 
  

   radius 
  unity, 
  

  

  where 
  both 
  triple 
  integrals 
  are 
  taken 
  throughout 
  the 
  trans- 
  

   formed 
  surface. 
  On 
  the 
  right 
  we 
  note 
  that 
  V, 
  as 
  indicated 
  

   by 
  the 
  stop, 
  acts 
  on 
  F{yp) 
  but 
  not 
  on 
  J(p). 
  In 
  general 
  V 
  

   acts 
  on 
  the 
  constituents 
  of 
  (/>. 
  The 
  propriety 
  of 
  the 
  inte- 
  

   grations 
  is 
  assumed. 
  In 
  case 
  <yp 
  is 
  a 
  homogeneous 
  function 
  

   of 
  p 
  of 
  degree 
  n, 
  ¥ 
  m 
  (yp) 
  is 
  homogeneous 
  of 
  degree 
  inn, 
  

   and 
  may 
  therefore 
  be 
  taken 
  as 
  any 
  homogeneous 
  function 
  

   which 
  gives 
  the 
  problem 
  a 
  meaning. 
  

  

  As 
  perhaps 
  the 
  simplest 
  illustration, 
  let 
  yp 
  = 
  (f)p, 
  a 
  self- 
  

   conjugate 
  linear 
  vector 
  function. 
  The 
  unit 
  sphere 
  is 
  carried 
  

   into 
  an 
  ellipsoid. 
  The 
  scalar 
  f(r) 
  becomes 
  a 
  function 
  whose 
  

   level 
  surfaces 
  are 
  similar 
  to 
  and 
  concentric 
  with 
  the 
  bounding 
  

   ellipsoid. 
  Since 
  w=l, 
  ¥ 
  m 
  (yp) 
  may 
  be 
  any 
  homogeneous 
  

   function 
  of 
  p 
  under 
  the 
  same 
  restrictions 
  of 
  continuity 
  as 
  in 
  

   Gauss's 
  theorem. 
  The 
  constant 
  Jacobian 
  cancels 
  out. 
  Hence 
  

  

  where 
  (j> 
  is 
  any 
  self-conjugate 
  linear 
  vector 
  function 
  with 
  

   constant 
  constituents, 
  and 
  the 
  triple 
  integrals 
  imply 
  inte- 
  

   gration 
  over 
  the 
  ellipsoid 
  T 
  2 
  <£/)=1. 
  

  

  2 
  K2 
  

  

  