﻿Prof. 
  D. 
  N. 
  Mallik 
  on 
  Fermat's 
  Law. 
  151 
  

  

  electrostatic 
  stresses 
  is 
  actually 
  that 
  given 
  by 
  Maxwell, 
  and 
  

   that 
  the 
  energy 
  due 
  to 
  it 
  is 
  localized 
  in 
  the 
  elements 
  of 
  

   volume 
  of 
  the 
  dielectric, 
  we 
  have, 
  in 
  an 
  element 
  of 
  volume 
  

   dF, 
  energy 
  "W, 
  where 
  

  

  and 
  therefore 
  

  

  *-£>. 
  

  

  •i7T 
  

  

  But 
  this 
  energy 
  must 
  be 
  equal 
  to 
  the 
  work 
  done 
  by 
  the 
  

   stresses 
  given 
  above. 
  

  

  To 
  find 
  an 
  expression 
  for 
  this, 
  let 
  <x 
  i 
  j3, 
  y 
  be 
  the 
  sides 
  of 
  

   the 
  volume 
  dl\ 
  so 
  that 
  a./3y 
  = 
  dT. 
  Then 
  the 
  increments 
  of 
  

   a. 
  /3. 
  y 
  due 
  to 
  the 
  electrostatic 
  stresses 
  will 
  be, 
  say, 
  ade 
  Xr 
  

   /3de 
  2 
  , 
  yde 
  3 
  . 
  And 
  the 
  work 
  done 
  by 
  the 
  stresses 
  will 
  be 
  

  

  /•"P 
  2 
  ATT 
  2 
  

  

  (!) 
  -qZ 
  ^7«^i 
  = 
  ~o~ 
  dTde 
  1 
  ; 
  

  

  8-7T 
  

  

  )TT 
  

  

  (2) 
  - 
  ={L 
  ,y 
  a 
  #fe 
  3 
  = 
  - 
  f- 
  </T,fe 
  s 
  ; 
  

  

  £F 
  2 
  £F 
  2 
  

  

  (3) 
  -ir 
  a 
  fo 
  de 
  *=---£ 
  Ldrde 
  *'> 
  

  

  07T 
  C7T 
  

  

  /•Fr/F 
  /-F 
  2 
  

  

  .-. 
  rfW= 
  — 
  ,- 
  £T= 
  — 
  rfrCrfex-^-rfea) 
  

  

  -±7T 
  07T 
  

  

  •'• 
  e 
  i 
  _ 
  € 
  2 
  — 
  e 
  3= 
  logF 
  2 
  + 
  const. 
  

   [Poincare, 
  Electricite 
  et 
  Optique.] 
  

  

  Now, 
  this 
  equation 
  cannot 
  obviously 
  be 
  satisfied, 
  if 
  we 
  

  

  regard 
  electrostatic 
  phenomena 
  as 
  involving 
  no 
  intrinsic 
  

  

  energy 
  (*. 
  e. 
  e 
  1 
  = 
  = 
  e 
  2 
  = 
  e 
  3 
  =F, 
  at 
  the 
  same 
  time); 
  but 
  if 
  we 
  

  

  can 
  postulate 
  a 
  certain 
  amount 
  of 
  intrinsic 
  energy, 
  we 
  are 
  

  

  able 
  to 
  do 
  so. 
  Thus, 
  if 
  we 
  suppose 
  

  

  F 
  2 
  

   62=62=63=0, 
  when 
  F 
  = 
  F 
  , 
  we 
  have 
  e 
  l 
  — 
  e 
  2 
  — 
  ^~ 
  log 
  " 
  .,, 
  

  

  -To 
  

  

  and 
  this 
  is 
  obviously 
  admissible 
  [with 
  a 
  further 
  limitation 
  on 
  

   the 
  arbitrariness 
  of 
  the 
  quantities 
  e 
  u 
  e 
  2 
  , 
  e 
  3 
  ]. 
  

  

  16. 
  All 
  these 
  theories 
  are 
  thus 
  seen 
  to 
  converge 
  to 
  the 
  

   same 
  point. 
  

  

  17. 
  Again, 
  from 
  the 
  principle 
  of 
  energy 
  

  

  T 
  + 
  V 
  = 
  C 
  (constant) 
  (3) 
  

  

  