﻿Prof. 
  D. 
  N. 
  Mallik 
  on 
  Fermat's 
  Law. 
  145 
  

  

  Then, 
  drawing 
  QQ'Q 
  ;/ 
  perpendicular 
  to 
  PQ 
  and 
  qq\ 
  RR' 
  

   perpendicular 
  to 
  QR, 
  we 
  have 
  

  

  since 
  /iPQ 
  + 
  ^'QR=/a'P^ 
  + 
  ^R, 
  

  

  or 
  yuQ"R'=//QR. 
  

  

  This 
  gives 
  Huyghens's 
  construction 
  for 
  singly 
  refracting 
  

   media. 
  

  

  4. 
  If 
  the 
  second 
  medium 
  is 
  doubly 
  refracting, 
  QQ', 
  RR' 
  

   are 
  the 
  traces 
  of 
  the 
  wave-fronts 
  (in 
  the 
  plane 
  of 
  the 
  paper) 
  

   and 
  thus, 
  since 
  the 
  wave-surface 
  is 
  the 
  envelope 
  of 
  the 
  wave- 
  

   front, 
  the 
  wave-surface 
  can 
  be 
  determined, 
  in 
  the 
  usual 
  way, 
  

   as 
  the 
  envelope 
  of 
  

  

  (Ix 
  + 
  my 
  + 
  nz) 
  = 
  v, 
  

  

  where 
  Z, 
  m, 
  n 
  are 
  the 
  direction-cosines 
  of 
  the 
  normal 
  to 
  the 
  

   wave-front 
  and 
  v 
  the 
  velocity 
  of 
  propagation, 
  while 
  /, 
  m, 
  n, 
  

   and 
  v 
  must 
  be 
  connected 
  by 
  a 
  relation 
  which, 
  however, 
  re- 
  

   quires 
  to 
  be 
  determined 
  on 
  some 
  theory, 
  independent 
  of 
  

   Fermat's 
  principle 
  (such 
  as 
  FresnePs). 
  Thus, 
  Fermat's 
  

   principle 
  is 
  seen 
  to 
  be 
  capable 
  of 
  giving 
  a 
  complete 
  kine- 
  

   matical 
  account 
  of 
  double 
  refraction, 
  also 
  — 
  with 
  a 
  subsidiary 
  

   hypothesis, 
  regarding 
  the 
  law 
  of 
  variation 
  of 
  fi 
  with 
  direction. 
  

  

  5. 
  Again, 
  since 
  <j}j 
  /j,ds 
  = 
  0,we 
  may 
  take 
  fj,ds 
  = 
  dY 
  y 
  and 
  we 
  

  

  have 
  §V 
  = 
  at 
  a 
  reflexion 
  or 
  refraction. 
  

   And, 
  since 
  

  

  dV 
  BV 
  ?)V 
  BV 
  

  

  <- 
  =fi, 
  or 
  ^— 
  .-3— 
  :^- 
  : 
  : 
  a 
  : 
  fi 
  : 
  y, 
  

   0-s 
  ^ 
  qx 
  oy 
  oz 
  ' 
  

  

  where 
  a, 
  /3, 
  y 
  are 
  the 
  direction-cosines 
  of 
  the 
  ray, 
  we 
  con- 
  

   clude 
  that 
  V 
  = 
  constant 
  is 
  a 
  surface 
  orthogonal 
  to 
  a 
  system 
  

   of 
  rays. 
  

  

  Phil 
  Mag. 
  S. 
  6. 
  Vol. 
  26. 
  No. 
  151. 
  July 
  1913. 
  L 
  

  

  