﻿Prof. 
  D. 
  N. 
  Mallik 
  on 
  Fermat's 
  Law. 
  147 
  

  

  Then 
  V 
  1 
  — 
  U 
  1 
  = 
  Vo 
  — 
  U 
  2 
  = 
  length 
  of 
  the 
  axial 
  ray 
  in 
  the 
  

   prism. 
  

  

  Let, 
  finally, 
  </>, 
  <f>' 
  be 
  the 
  angles 
  of 
  incidence 
  and 
  refraction 
  

   at 
  the 
  first 
  surface, 
  and 
  

  

  'i/r, 
  yjr' 
  be 
  the 
  angles 
  of 
  incidence 
  and 
  refraction 
  at 
  the 
  

   second 
  surface. 
  

  

  The 
  edge 
  of 
  the 
  prism 
  being 
  taken 
  as 
  the 
  axis 
  of 
  y 
  and 
  

   the 
  normal 
  to 
  the 
  face 
  of 
  incidence 
  as 
  the 
  axis 
  of 
  z, 
  the 
  

   direction-cosines 
  of 
  one 
  of 
  the 
  focal 
  lines 
  before 
  incidence 
  

   are 
  : 
  

  

  sin 
  a 
  cos 
  c/>, 
  cos 
  a, 
  sin 
  a 
  sin 
  (j). 
  

  

  Thus, 
  the 
  characteristic 
  function 
  before 
  incidence 
  becomes 
  

  

  X 
  = 
  /jl{cc 
  sin 
  4> 
  + 
  z 
  cos 
  (f> 
  — 
  - 
  — 
  (,i'smacos<£ 
  + 
  v/cosa 
  + 
  2sinasin<£) 
  2 
  

  

  — 
  — 
  (a;cosacos</>— 
  ysina 
  + 
  ^cosa 
  sin<£) 
  2 
  }. 
  

  

  From 
  the 
  continuity 
  of 
  the 
  function, 
  at 
  z=0, 
  we 
  get 
  by 
  

   equating 
  coefficients 
  of 
  .1*, 
  <Z" 
  2 
  , 
  xy, 
  ?/ 
  2 
  , 
  

  

  ja 
  sin 
  c/> 
  = 
  ft' 
  sin 
  <f>' 
  

   /sin 
  2 
  a 
  cos 
  2 
  a\ 
  0J 
  ,/sin 
  2 
  /3 
  cos 
  2 
  /3\ 
  

  

  /cos 
  2 
  a 
  sin 
  2 
  a\ 
  /cos 
  2 
  # 
  sin 
  2 
  /3 
  \ 
  

  

  //, 
  sin 
  a 
  cos 
  a 
  [ 
  ) 
  cos 
  <j> 
  = 
  p 
  sin 
  /3 
  cos 
  /3 
  [ 
  -pr 
  — 
  |y- 
  )cos<£/, 
  

  

  and 
  similar 
  equations 
  for 
  the 
  second 
  refraction. 
  

  

  9. 
  In 
  order 
  to 
  take 
  account 
  of 
  aberration, 
  we 
  must 
  obtain 
  

   the 
  characteristic 
  function 
  up 
  to 
  the 
  order 
  zx 
  2 
  . 
  

  

  Let 
  the 
  equation 
  of 
  the 
  characteristic 
  surface 
  be 
  

  

  2 
  2 
  

  

  2z= 
  - 
  + 
  *- 
  + 
  2z{*x 
  2 
  + 
  fay 
  + 
  yy 
  2 
  ) 
  + 
  &c. 
  

  

  Pi 
  P2 
  

  

  Now 
  the 
  perpendicular 
  from 
  the 
  origin 
  on 
  the 
  tangent 
  plane 
  

  

  x 
  1/ 
  

  

  at 
  .'■'. 
  v 
  ', 
  z' 
  to 
  the 
  surface 
  2z= 
  '- 
  — 
  I- 
  — 
  is 
  

   9 
  Pi 
  Pi 
  

  

  v 
  pr 
  p 
  2 
  2 
  

  

  if 
  terms 
  of 
  the 
  order 
  zx 
  2 
  are 
  retained. 
  

  

  L2 
  

  

  