﻿Combination 
  witli 
  Homogeneous 
  Functions. 
  465 
  

  

  into 
  which 
  the 
  given 
  surface 
  is 
  carried 
  by 
  the 
  transformation 
  

   p 
  = 
  X 
  n 
  {p'), 
  or 
  simply 
  Xp 
  f 
  . 
  We 
  note 
  that 
  /\, 
  as 
  indicated 
  by 
  

   the 
  brackets, 
  operates 
  on 
  J' 
  as 
  well 
  as 
  F. 
  To 
  take 
  an 
  

   example, 
  the 
  problem 
  worked 
  at 
  the 
  close 
  of 
  Art. 
  4 
  would 
  

   read 
  by 
  the 
  present 
  method 
  

  

  Jj 
  ]V+/)T 
  w 
  rfV 
  = 
  \ 
  f/dr 
  .^\A(aV 
  + 
  bY-)abcdY', 
  

  

  where 
  the 
  right-hand 
  integral 
  is 
  over 
  a 
  unit 
  sphere, 
  leading 
  

   to 
  the 
  same 
  answer 
  as 
  before. 
  Here 
  Xp' 
  ' 
  = 
  iax' 
  +jby' 
  + 
  kcz\ 
  

   so 
  that 
  the 
  transformation 
  is 
  equivalent 
  to 
  putting 
  ax' 
  

   for 
  x, 
  &c. 
  

  

  6. 
  Analogue 
  in 
  two 
  dimensions. 
  — 
  By 
  parallel 
  reasoning 
  we 
  

   may 
  work 
  problems 
  of 
  the 
  same 
  cha'racter 
  in 
  a 
  plane. 
  Thus 
  if 
  

   F 
  be 
  homogeneous 
  in 
  a 
  plane 
  point-vector 
  p 
  } 
  that 
  is 
  in 
  x 
  and 
  ?/, 
  

  

  and 
  if 
  A=^— 
  o 
  + 
  ^- 
  , 
  

   Ox 
  2 
  ^y- 
  

  

  For 
  inula 
  (B) 
  

  

  $/« 
  • 
  K(p)dA= 
  -L^y+]f(r)dr 
  . 
  j"jAF 
  m 
  (p)rfA, 
  . 
  (B) 
  

  

  where 
  the 
  double 
  integrals 
  are 
  to 
  be 
  laken 
  over 
  the 
  area 
  of 
  a 
  

   circle 
  of 
  radius 
  a 
  with 
  centre 
  at 
  the 
  origin 
  of 
  p. 
  

  

  As 
  an 
  example 
  easy 
  to 
  verify 
  by 
  other 
  methods, 
  let 
  it 
  be 
  

  

  CCx 
  2 
  +v 
  2 
  

  

  required 
  to 
  carry 
  the 
  integral 
  l 
  j 
  — 
  — 
  2- 
  dA 
  over 
  the 
  area 
  

  

  included 
  within 
  the 
  circle 
  x 
  2 
  -+- 
  y- 
  = 
  2ax. 
  The 
  transfor- 
  

   mation 
  x 
  = 
  x' 
  2 
  , 
  y=x'y\ 
  carries 
  the 
  area 
  within 
  the 
  circle 
  into 
  

   the 
  area 
  included 
  between 
  the 
  axis 
  of 
  X 
  / 
  and 
  the 
  new 
  

   circle 
  x' 
  2 
  + 
  y' 
  2 
  = 
  2a, 
  on 
  either 
  side 
  of 
  OX'. 
  The 
  Jacobian 
  

  

  is 
  2x' 
  2 
  . 
  The 
  transformed 
  integral 
  is 
  I 
  l 
  — 
  x' 
  2 
  dA', 
  which 
  if 
  

  

  m* 
  

  

  taken 
  over 
  the 
  whole 
  of 
  the 
  new 
  circle 
  gives 
  twice 
  the 
  

   original 
  quantity. 
  ~\Ye 
  therefore 
  have, 
  dropping 
  accents 
  and 
  

   applying 
  (B), 
  (remembering 
  that 
  the 
  new 
  radius 
  is 
  V2a), 
  

  

  1 
  1 
  , 
  ^ 
  

  

  o 
  

  

  the 
  double 
  integral 
  being 
  over 
  the 
  whole 
  circle. 
  

  

  7. 
  Application 
  to 
  a 
  hemisphere.— 
  -By 
  the 
  combined 
  use 
  of 
  

   (A) 
  and 
  (B) 
  we 
  may 
  obtain 
  a 
  reduction 
  process 
  for 
  inte- 
  

   grating 
  a 
  homogeneous 
  function 
  over 
  a 
  hemispherical 
  volume. 
  

   The 
  relation 
  (5) 
  is 
  unchanged 
  in 
  form, 
  but 
  we 
  now 
  carry 
  the 
  

  

  