﻿464 
  Dr. 
  F. 
  L. 
  Hitchcock 
  on 
  the 
  Operator 
  V 
  in 
  

  

  To 
  take 
  a 
  specific 
  example, 
  let 
  us 
  extend 
  to 
  the 
  volume 
  of 
  the 
  

   ellipsoid 
  the 
  problem 
  worked 
  at 
  the 
  close 
  of 
  Art. 
  2, 
  the 
  equation 
  

  

  of 
  the 
  surface 
  being 
  -£ 
  + 
  |! 
  + 
  * 
  = 
  1, 
  the 
  density 
  equal 
  to 
  

  

  the 
  square 
  root 
  of 
  the 
  left 
  side 
  of 
  this 
  equation, 
  and 
  the 
  moment 
  

   of 
  inertia 
  about 
  OZ 
  required. 
  The 
  transformation 
  is 
  given 
  

  

  b 
  J 
  <f>p 
  = 
  i 
  ~+jj 
  4- 
  k~. 
  The 
  Laplacean 
  operator 
  A 
  trans- 
  

  

  a 
  C 
  

  

  forms 
  into 
  — 
  (0- 
  1 
  V) 
  2 
  , 
  that 
  is 
  into 
  

  

  a? 
  — 
  4- 
  h 
  2 
  — 
  -J- 
  r 
  2 
  ^ 
  

   and 
  the 
  transformed 
  equation 
  reads 
  

  

  where 
  the 
  volume 
  integrals 
  are 
  taken 
  over 
  the 
  volume 
  of 
  

   the 
  ellipsoid. 
  The 
  right 
  side 
  by 
  inspection 
  is 
  i\N(a 
  2 
  + 
  b 
  2 
  )dY 
  

   ov%7r(a 
  2 
  + 
  h 
  2 
  )abc. 
  

  

  5. 
  Inverse 
  method 
  of 
  transformation. 
  — 
  When 
  a 
  surface 
  is 
  

   given, 
  and 
  when 
  we 
  can 
  transform 
  it 
  into 
  a 
  sphere, 
  we 
  can 
  

   apply 
  (A) 
  direct! 
  j 
  if 
  the 
  integrand 
  is 
  homogeneous 
  after 
  the 
  

   transformation. 
  The 
  transformation 
  is 
  then 
  the 
  inverse 
  of 
  

   7/3 
  in 
  the 
  last 
  article. 
  This 
  method, 
  while 
  resting 
  on 
  the 
  

   same 
  principles 
  as 
  the 
  foregoing, 
  differs 
  in 
  using 
  an 
  untrans- 
  

   formed 
  A- 
  To 
  illustrate, 
  let 
  p 
  = 
  \(p') 
  = 
  y-\p') 
  be 
  the 
  trans- 
  

   formation 
  which 
  carries 
  a 
  given 
  surface 
  into 
  a 
  sphere. 
  Let 
  

   the 
  Jacobian 
  of 
  the 
  transformation 
  X 
  be 
  J', 
  and 
  let 
  A, 
  be 
  

   homogeneous 
  of 
  degree 
  n. 
  We 
  then 
  obtain 
  by 
  (A) 
  

  

  ^/(lyp) 
  .V 
  m 
  (p)dV 
  

  

  '/) 
  . 
  F„0„p) 
  . 
  J'(p')dV, 
  by 
  the 
  trf. 
  p= 
  V, 
  

  

  where 
  p 
  = 
  mn-f 
  3(n 
  — 
  1), 
  the 
  degree 
  of 
  the 
  integrand 
  after 
  

   the 
  transformation. 
  The 
  integration 
  on 
  the 
  left 
  is 
  carried 
  

   over 
  the 
  volume 
  of 
  the 
  original 
  surface, 
  the 
  others 
  are 
  over 
  

   a 
  sphere 
  of 
  radius 
  a 
  and 
  centre 
  at 
  the 
  origin 
  of 
  coordinates 
  

  

  