﻿Vector 
  Differentials. 
  577 
  

  

  The 
  mathematical 
  connexion 
  between 
  these 
  two 
  kinds 
  of 
  

   magnitudes 
  is 
  extremely 
  intimate: 
  if 
  we 
  have 
  any 
  scalar 
  

   function 
  continuously 
  distributed 
  through 
  a 
  portion 
  of 
  space, 
  

   there 
  is 
  a 
  vector 
  function 
  immediately 
  derivable 
  from 
  it 
  by 
  

   the 
  operator 
  V? 
  which 
  derived 
  vector 
  was 
  called 
  by 
  Maxwell 
  

   the 
  space-variation 
  of 
  the 
  original 
  scalar. 
  

  

  The 
  object 
  of 
  the 
  present 
  paper 
  is 
  to 
  study 
  briefly 
  the 
  

   differentiation 
  of 
  vectors, 
  a 
  subject 
  inseparably 
  bound 
  up 
  with 
  

   the 
  quaternion 
  operators 
  V 
  an( 
  l 
  <£. 
  I 
  shall 
  assume 
  that 
  the 
  

   reader 
  has 
  some 
  slight 
  acquaintance 
  with 
  the 
  calculus 
  of 
  

   Hamilton, 
  and 
  shall 
  occasionally 
  refer 
  to 
  Tait's 
  ' 
  Quaternions/ 
  

   3rd 
  edition, 
  1890. 
  

  

  2. 
  From 
  the 
  definition 
  

  

  • 
  rf(F,)=-»{ 
  F 
  ( 
  S+ 
  f)-F,} 
  

  

  follows 
  the 
  very 
  general 
  proposition 
  that 
  a 
  differential 
  is 
  a 
  

   linear 
  function 
  : 
  both 
  q 
  and 
  F^ 
  are, 
  in 
  general, 
  quaternions 
  ; 
  

   but 
  one 
  or 
  both 
  habitually 
  " 
  degenerate 
  " 
  into 
  vectors 
  or 
  

   scalars. 
  In 
  any 
  case 
  d(¥q) 
  is 
  linear 
  in 
  dq. 
  

  

  It 
  follows 
  that 
  if 
  P 
  is 
  any 
  scalar 
  function 
  of 
  a 
  point 
  p, 
  then 
  

   dF 
  is 
  linear 
  in 
  dp. 
  

  

  Now 
  every 
  possible 
  scalar 
  term 
  linear 
  in 
  dp 
  may, 
  by 
  very 
  

   elementary 
  transformations, 
  be 
  put 
  in 
  the 
  form 
  $\dp, 
  where 
  

   X 
  is 
  of 
  course 
  a 
  vector 
  function 
  of 
  p. 
  If 
  there 
  are 
  several 
  

   such 
  terms 
  we 
  may 
  assume 
  that 
  2\= 
  — 
  VP> 
  where 
  the 
  minus 
  

   sign 
  is 
  introduced 
  in 
  order 
  that 
  our 
  results 
  may 
  agree 
  with 
  

   Hamilton's 
  original 
  definition 
  of 
  V- 
  Therefore 
  

  

  rfP=-Sd/oVP, 
  (1) 
  

  

  which 
  is 
  a 
  fundamental 
  equation. 
  

  

  From 
  this, 
  remembering 
  that 
  dp 
  may, 
  like 
  any 
  other 
  vector, 
  

   be 
  thought 
  of 
  as 
  the 
  product 
  of 
  a 
  scalar 
  and 
  a 
  unit-vector, 
  we 
  

   have 
  

  

  S 
  = 
  -8eVP, 
  (2) 
  

  

  6 
  ' 
  

  

  by 
  writing 
  dp 
  = 
  edh 
  and 
  then 
  dividing 
  both 
  sides 
  by 
  dh. 
  

  

  Here 
  -== 
  may 
  be 
  thought 
  of 
  as 
  an 
  operator. 
  It 
  signifies 
  

  

  differentiation 
  with 
  regard 
  to 
  any 
  direction 
  whatever 
  in 
  

   space, 
  and 
  e 
  is 
  the 
  corresponding 
  unit-vector, 
  either 
  a 
  con- 
  

   stant 
  or 
  a 
  function 
  of 
  p. 
  

  

  