﻿Vector 
  Differentials. 
  581 
  

  

  whence 
  by 
  putting 
  fidhp 
  for 
  dp, 
  

  

  $:- 
  B 
  *-"^ 
  • 
  • 
  • 
  (11) 
  

  

  where 
  a 
  and 
  /3 
  are 
  perfectly 
  interchangeable, 
  because 
  either 
  

   of 
  them 
  is 
  any 
  constant 
  unit-vector 
  whatever. 
  

   One 
  consequence 
  is 
  that 
  

  

  v£=*«=ivp, 
  • 
  • 
  • 
  • 
  (12) 
  

  

  which 
  may 
  be 
  extended 
  to 
  a 
  vector 
  by 
  the 
  usual 
  method 
  

   (Tait, 
  § 
  149). 
  Thus 
  the 
  operators 
  V 
  and 
  jr 
  are 
  commu- 
  

   tative, 
  provided 
  the 
  direction 
  h 
  is 
  constant. 
  A 
  single 
  case 
  of 
  

   the 
  same 
  kind 
  will 
  presently 
  be 
  exhibited 
  where 
  the 
  direction 
  

   of 
  differentiation 
  is 
  not 
  constant. 
  

  

  6. 
  The 
  function 
  ^, 
  found 
  by 
  differentiating 
  TJVP, 
  or 
  v, 
  

   owes 
  most 
  of 
  its 
  peculiarities 
  to 
  the 
  fact 
  that 
  the 
  differential 
  

   of 
  a 
  unit-vector 
  is 
  always 
  at 
  right 
  angles 
  to 
  the 
  unit-vector 
  

   itself 
  (Tait, 
  § 
  140, 
  (2)) 
  ; 
  this 
  is 
  expressed 
  by 
  the 
  equation 
  

  

  S^e 
  = 
  0, 
  (13) 
  

  

  where 
  e 
  is 
  any 
  direction 
  whatever. 
  Thus 
  the 
  strain 
  y 
  turns 
  

   every 
  vector 
  into 
  the 
  tangent 
  plane 
  to 
  the 
  surface 
  P 
  = 
  const. 
  

   If 
  we 
  form 
  the 
  strain-cubic 
  in 
  the 
  usual 
  manner 
  we 
  find 
  

   that 
  the 
  absolute 
  term 
  vanishes, 
  so 
  that 
  

  

  X(% 
  2 
  — 
  w? 
  2% 
  + 
  m 
  1 
  )=0 
  

  

  for 
  any 
  direction 
  whatever. 
  Thus 
  the 
  cubic 
  has 
  a 
  zero 
  root 
  ; 
  

   for 
  another 
  way 
  of 
  finding 
  it 
  we 
  have, 
  X 
  and 
  fi 
  being 
  any 
  

   vectors 
  whatever, 
  

  

  %Vx'^> 
  = 
  0, 
  Tait, 
  § 
  157, 
  (2). 
  

  

  By 
  interchanging 
  y 
  and 
  y' 
  in 
  this 
  last 
  equation 
  we 
  obtain 
  

  

  %V=0, 
  (14) 
  

  

  for 
  V%\%/u, 
  is 
  parallel 
  to 
  v, 
  by 
  (IB). 
  It 
  appears 
  from 
  these 
  

   results 
  that 
  that 
  direction 
  for 
  which 
  % 
  = 
  is 
  at 
  right 
  angles 
  

   to 
  the 
  plane 
  into 
  which 
  y' 
  turns 
  every 
  vector 
  ; 
  and 
  vice 
  versa. 
  

   Whence, 
  by 
  taking 
  a 
  special 
  case 
  of 
  (8), 
  

  

  (X-X> 
  = 
  V(Wv>, 
  

   the 
  left 
  side 
  reduces 
  to 
  yv, 
  that 
  is, 
  -*-; 
  and 
  remembering 
  

   that 
  v 
  satisfies 
  the 
  equation 
  SvVv 
  = 
  0, 
  we 
  see 
  that 
  v, 
  yv, 
  and 
  

  

  