﻿Prof. 
  D. 
  N. 
  Mallik 
  on 
  Fer 
  mat's 
  Law. 
  149 
  

  

  We 
  conclude, 
  therefore, 
  that 
  all 
  the 
  kinematical 
  results 
  

   of 
  optics 
  can 
  be 
  deduced 
  from 
  Fermat's 
  law. 
  

  

  10. 
  Proceeding, 
  now, 
  to 
  the 
  dynamical 
  significance 
  of 
  

   Fermat's 
  law 
  (which 
  Fermat 
  deduced 
  from 
  metaphysical 
  

   considerations), 
  we 
  observe 
  that 
  the 
  configuration 
  of 
  equi- 
  

   librium 
  and 
  motion 
  of 
  a 
  dynamical 
  system 
  is 
  defined 
  by 
  

   S$(T-Y)dt=0, 
  where 
  

  

  T 
  = 
  Kinetic 
  energy, 
  

   V 
  = 
  Potential 
  energy. 
  

  

  If 
  this 
  is 
  to 
  be 
  consistent 
  with 
  Fermat's 
  law 
  we 
  must 
  have, 
  

   for 
  light 
  propagation, 
  

  

  T-V=C 
  (constant) 
  (1) 
  

  

  Also, 
  if 
  Y' 
  = 
  Potential 
  energy 
  of 
  light 
  disturbance 
  during 
  

   the 
  displacement 
  of 
  the 
  disturbance 
  through 
  the 
  length 
  of 
  a 
  

   wave, 
  

  

  §Tdt=$V'dt 
  (2) 
  

  

  From 
  (1) 
  and 
  (2) 
  we 
  get 
  

  

  v+c=v, 
  

  

  showing 
  that 
  we 
  must 
  postulate 
  a 
  certain 
  intrinsic 
  energy 
  in 
  

   the 
  system, 
  whose 
  motion 
  is 
  concerned 
  with 
  the 
  propagation, 
  

   in 
  order 
  to 
  justify 
  the 
  principle 
  of 
  swiftest 
  propagation. 
  

  

  11. 
  Now, 
  all 
  the 
  optical 
  theories 
  which 
  have 
  been 
  at 
  all 
  

   successful 
  in 
  giving 
  a 
  fairly 
  satisfactory 
  account 
  of 
  the 
  

   mechanism 
  of 
  light 
  propagation 
  depend 
  on 
  the 
  assumption 
  

   that 
  the 
  potential 
  energy 
  of 
  light 
  disturbance 
  in 
  a 
  homo- 
  

   geneous 
  medium 
  is 
  of 
  the 
  form 
  

  

  jj 
  J 
  ( 
  w 
  * 
  2 
  + 
  w 
  y~ 
  + 
  w 
  z 
  2 
  ) 
  dxdydz. 
  

  

  Thus, 
  on 
  McCullagh's 
  theory, 
  w 
  XJ 
  w 
  y 
  , 
  iv 
  z 
  are 
  the 
  molecular 
  

   rotations 
  of 
  the 
  elastic 
  medium, 
  behaving 
  as 
  an 
  elastic 
  solid. 
  

  

  On 
  the 
  labile 
  aether 
  theory 
  also, 
  the 
  energy 
  function 
  is 
  

   found 
  to 
  assume 
  the 
  same 
  form, 
  together 
  with 
  certain 
  surface 
  

   integrals 
  which 
  vanish 
  on 
  account 
  of 
  the 
  " 
  labileness 
  " 
  of 
  the 
  

   medium 
  at 
  the 
  boundaries. 
  

  

  And, 
  finally, 
  on 
  the 
  electromagnetic 
  theory, 
  w 
  Xi 
  w 
  y 
  , 
  w 
  2 
  are 
  

   the 
  electric 
  displacements 
  of 
  Maxwell. 
  

  

  12. 
  It 
  will 
  be 
  seen, 
  however, 
  that 
  if 
  the 
  above 
  expression 
  

   is 
  to 
  be 
  true 
  of 
  each 
  element 
  of 
  volume, 
  the 
  equilibrium 
  

   condition 
  fas 
  was 
  pointed 
  out 
  by 
  Stokes) 
  is 
  not 
  satisfied, 
  for 
  

   it 
  makes 
  «? 
  y 
  = 
  —yx, 
  where 
  x 
  yj 
  y 
  x 
  are 
  the 
  tangential 
  stresses 
  on 
  

   the 
  opposite 
  faces 
  of 
  an 
  elementary 
  parallelopiped. 
  Now 
  

  

  