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  LIV. 
  On 
  the 
  Operator 
  V 
  in 
  Combination 
  with 
  Homogeneous 
  

   Functions. 
  Second 
  Paper. 
  By 
  Frank 
  L. 
  Hitchcock, 
  

   Massachusetts 
  Institute 
  of 
  Technology, 
  Cambridge 
  , 
  Mass.* 
  

  

  I. 
  INTRODUCTION.— 
  In 
  a 
  former 
  paper 
  (Phil. 
  Mag. 
  

   vol. 
  xxix. 
  May 
  1915, 
  p. 
  700), 
  two 
  theorems 
  were 
  

   developed, 
  which 
  yield 
  properties 
  of 
  any 
  homogeneous 
  vector 
  

   in 
  connexion 
  with 
  the 
  operators 
  SV 
  and 
  VV- 
  I 
  now 
  pro- 
  

   pose 
  to 
  study 
  in 
  a 
  somewhat 
  similar 
  manner 
  the 
  Laplacean 
  

   operator 
  A, 
  or 
  — 
  V 
  2 
  , 
  defined 
  by 
  

  

  H& 
  + 
  & 
  + 
  £ 
  (1) 
  

  

  in 
  three 
  dimensions 
  (with 
  analogous 
  definition 
  for 
  two 
  di- 
  

   mensions). 
  The 
  results 
  obtained 
  are 
  of 
  two 
  sorts. 
  First, 
  as 
  

   a 
  more 
  elementary 
  matter, 
  I 
  shall 
  show 
  how 
  a 
  large 
  number 
  

   of 
  reduction 
  formulas 
  may 
  be 
  written 
  down, 
  useful 
  in 
  the 
  

   evaluation 
  of 
  multiple 
  integrals. 
  Later 
  I 
  shall 
  prove 
  a 
  

   theorem 
  similar 
  to 
  those 
  of 
  the 
  first 
  paper, 
  giving 
  the 
  form 
  

   of 
  the 
  function 
  by 
  which 
  various 
  homogeneous 
  scalars 
  or 
  

   vectors 
  differ 
  from 
  harmonic 
  functions 
  of 
  the 
  same 
  degree. 
  

  

  2. 
  The 
  fundamental 
  reduction 
  formula. 
  — 
  Let 
  us 
  first 
  take, 
  

   as 
  in 
  the 
  former 
  paper, 
  F(p) 
  a 
  function 
  of 
  the 
  point-vector 
  p, 
  

   homogeneous 
  of 
  degree 
  m. 
  In 
  this 
  case, 
  however, 
  T 
  restrict 
  

   m 
  to 
  be 
  positive, 
  and 
  shall 
  at 
  first 
  suppose 
  F(p) 
  to 
  be 
  a 
  scalar 
  

   function. 
  Write 
  r=Tp, 
  and 
  let 
  f(r) 
  be 
  a 
  scalar 
  function 
  

   of 
  r. 
  Consider 
  the 
  problem 
  of 
  integrating 
  the 
  product 
  

   f(r).¥(p'} 
  over 
  the 
  volume 
  of 
  a 
  sphere 
  of 
  radius 
  a 
  with 
  

   centre 
  at 
  the 
  origin 
  of 
  coordinates 
  : 
  I 
  shall 
  prove 
  the 
  

   following 
  reduction 
  formula, 
  — 
  

  

  Formula 
  (A) 
  

   ^if(r).F,Ap)dV=^X/'' 
  + 
  V(>^r-^AFJ 
  P 
  )dV, 
  (A) 
  

  

  where 
  the 
  triple 
  integrations 
  are 
  to 
  be 
  taken 
  over 
  the 
  volume 
  

   of 
  the 
  sphere. 
  We 
  begin 
  with 
  Gauss's 
  theorem, 
  

  

  where 
  X, 
  Y, 
  and 
  Z 
  are 
  scalars, 
  I, 
  m, 
  and 
  n 
  are 
  the 
  direction 
  

   cosines 
  of 
  the 
  ouiward 
  normal 
  to 
  a 
  closed 
  surface, 
  and 
  the 
  

   integrations 
  are 
  carried 
  over 
  the 
  surface 
  and 
  throughout 
  its 
  

  

  * 
  Communicated 
  by 
  the 
  Author. 
  

   Phil. 
  Mag. 
  S. 
  6, 
  Vol. 
  35. 
  No. 
  210. 
  June 
  1918, 
  2 
  K 
  

  

  