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  14* 
  ] 
  

  

  V. 
  Format's 
  Law. 
  By 
  Prof. 
  D. 
  N. 
  Mallik, 
  

   B.A., 
  Sc.D., 
  FJR.S.E* 
  

  

  1. 
  TT^EIIMAT'S 
  law 
  of 
  swiftest 
  propagation 
  of 
  light 
  is 
  

   .1- 
  3§dt 
  = 
  0, 
  where 
  t 
  is 
  the 
  time 
  of 
  propagation 
  and 
  8 
  

   is 
  the 
  operator 
  of 
  the 
  calculus 
  of 
  variation. 
  Now, 
  since 
  on 
  

   the 
  wave-theory 
  jn 
  the 
  index 
  of 
  refraction 
  varies 
  inversely 
  

   as 
  the 
  velocity 
  of 
  propagation, 
  the 
  above 
  is 
  obviously 
  the 
  

   same 
  as 
  8\ 
  /j,ds 
  = 
  0, 
  an 
  equation 
  which, 
  as 
  is 
  well 
  known, 
  

   analytically 
  embodies 
  a 
  complete 
  kinematical 
  statement 
  of 
  

   ail 
  optical 
  phenomena. 
  

   2. 
  Thus, 
  

  

  8 
  \ 
  fids 
  - 
  \ 
  8fids-}~ 
  \ 
  fiid8s 
  

  

  since 
  d$ 
  2 
  = 
  dx 
  2 
  + 
  dy 
  2 
  -\-dz 
  2 
  , 
  

  

  and 
  therefore 
  7 
  . 
  fdx 
  7 
  ~ 
  \ 
  

  

  Therefore, 
  integrating 
  by 
  parts, 
  we 
  have 
  obviously, 
  

  

  or 
  15/jL 
  _ 
  d 
  / 
  dx\ 
  * 
  

  

  "fix 
  ~~ 
  ds\ 
  ds 
  / 
  

  

  d/v 
  

   while, 
  at 
  a 
  bounding 
  surface, 
  /JL^- 
  y 
  &c, 
  are 
  continuous, 
  

  

  giving 
  the 
  laws 
  of 
  reflexion 
  and 
  refraction 
  and 
  the 
  path 
  of 
  a 
  

   ray 
  in 
  a 
  heterogeneous 
  singly-refracting 
  medium. 
  

  

  3. 
  Again, 
  let 
  PQ, 
  QR 
  be 
  the 
  incident 
  and 
  refracted 
  rays 
  

   at 
  a 
  surface 
  of 
  separation 
  between 
  the 
  media 
  of 
  refractive 
  

   indices 
  p, 
  ///, 
  and 
  let 
  P'B/ 
  be 
  a 
  ray 
  parallel 
  to 
  PQ, 
  and 
  P</R 
  

   a 
  ray 
  consecutive 
  to 
  PQR. 
  

  

  * 
  Communicated 
  "by 
  the 
  Author. 
  An 
  abstract 
  of 
  this 
  paper 
  was 
  read 
  

   at 
  the 
  Annual 
  General 
  Meeting 
  of 
  the 
  Indian 
  Association 
  for 
  the 
  

   Cultivation 
  of 
  Science, 
  Calcutta, 
  held 
  on 
  Thursday, 
  the 
  28th 
  November, 
  

   1912. 
  First 
  published 
  as 
  a 
  Bulletin 
  of 
  that 
  Association. 
  

  

  