﻿Vector 
  Differentials. 
  583 
  

  

  end 
  of 
  this 
  paper. 
  But 
  we 
  are 
  now 
  concerned 
  to 
  get 
  an 
  

   expression 
  for 
  V(^)* 
  -^ 
  ^ 
  s 
  P 
  r 
  °ved 
  above 
  that 
  

  

  tydp 
  == 
  — 
  — 
  vSvdp 
  — 
  ti/$xvdp 
  + 
  tydp, 
  

  

  where, 
  by 
  Art. 
  8, 
  the 
  first 
  two 
  terms 
  give 
  — 
  -5- 
  and 
  — 
  tv%y 
  9 
  

  

  and 
  the 
  last 
  term 
  gives 
  t\J 
  v 
  - 
  Thus 
  if 
  P 
  satisfies 
  Laplace's 
  

  

  equation, 
  then 
  

  

  dt 
  r^ 
  

  

  -^-tv 
  X 
  v 
  + 
  tVv=0; 
  

  

  the 
  vector 
  part 
  gives 
  an 
  independent 
  proof 
  of 
  (15) 
  ; 
  the 
  scalar, 
  

   part 
  is 
  

  

  dt 
  ,<*-■ 
  

  

  and 
  since 
  it 
  has 
  already 
  been 
  proved 
  that, 
  in 
  general, 
  

   d(tv) 
  dt 
  

  

  **--%?=* 
  sit*** 
  

  

  we 
  have, 
  provided 
  P 
  satisfies 
  Laplace's 
  equation, 
  

  

  V* 
  = 
  *(vSVv 
  + 
  w) 
  (17) 
  

  

  The 
  vector 
  vSVv-r 
  x 
  v 
  ma 
  y 
  ^ 
  e 
  written 
  Vv 
  . 
  v 
  ; 
  and 
  because 
  

   V 
  2 
  t 
  is 
  a 
  scalar, 
  

  

  W(*Vf.f)=0 
  

  

  which 
  reduces 
  at 
  once 
  to 
  

  

  VV(Vv.v) 
  = 
  0; 
  

   from 
  (10), 
  putting 
  Vv 
  for 
  7 
  and 
  v 
  for 
  r 
  and 
  taking 
  vectors, 
  

  

  V(VVV-V(Vv) 
  2 
  + 
  2SVv 
  . 
  VVv-2 
  x 
  VVv=0, 
  

  

  where 
  the 
  second 
  and 
  third 
  terms 
  destroy 
  each 
  other, 
  so 
  that 
  

   finally 
  

  

  VvV 
  2 
  v 
  + 
  2 
  x 
  VVv 
  = 
  0, 
  .... 
  (18 
  a) 
  

  

  which 
  is 
  the 
  required 
  condition. 
  

  

  The 
  same 
  essential 
  fact 
  is 
  expressed 
  by 
  saying 
  that 
  Vv 
  . 
  v 
  

   must 
  be 
  integrable 
  without 
  a 
  factor, 
  or 
  that 
  there 
  must 
  exist 
  a 
  

   scalar 
  — 
  call 
  it 
  u 
  — 
  such 
  that 
  

  

  u=V^ 
  l 
  {Vv 
  . 
  1/). 
  ...... 
  (19) 
  

  

  8. 
  To 
  examine 
  the 
  properties 
  of 
  X^Vv, 
  we 
  may 
  write, 
  as 
  a 
  

  

  