﻿140 
  Prof. 
  D. 
  N. 
  Mallik 
  on 
  Fermafs 
  Law. 
  

  

  Accordingly, 
  if 
  a 
  given 
  surface 
  can, 
  at 
  any 
  stage, 
  be 
  made 
  

   to 
  coincide 
  with 
  a 
  surface 
  Y 
  = 
  constant, 
  a 
  surface 
  can 
  always 
  

   be 
  drawn 
  to 
  coincide 
  with 
  any 
  other 
  member 
  of 
  the 
  family 
  

   V 
  = 
  constant 
  : 
  or 
  

  

  Any 
  system 
  of 
  rays 
  originally 
  orthogonal 
  to 
  a 
  surface 
  

   will 
  always 
  be 
  orthogonal 
  to 
  a 
  surface, 
  after 
  any 
  number 
  of 
  

   reflexions 
  and 
  refractions. 
  

  

  6. 
  Since, 
  then, 
  the 
  rays 
  of 
  light 
  (which 
  are 
  orthogonal 
  to 
  

   ~a 
  surface) 
  may 
  be 
  regarded 
  as 
  normals 
  to 
  a 
  family 
  of 
  sur- 
  

   faces, 
  and 
  by 
  Sturm's 
  theorem 
  all 
  the 
  normals 
  to 
  a 
  surface 
  

   in 
  the 
  neighbourhood 
  of 
  a 
  poiut 
  converge 
  to 
  or 
  diverge 
  from 
  

   two 
  focal 
  lines 
  at 
  right 
  angles 
  to 
  one 
  another, 
  each 
  of 
  which 
  

   passes 
  through 
  the 
  centre 
  of 
  curvature 
  of 
  one 
  of 
  the 
  prin- 
  

   cipal 
  normal 
  sections 
  and 
  is 
  perpendicular 
  to 
  the 
  plane 
  of 
  

   that 
  section, 
  we 
  conclude 
  that 
  all 
  the 
  rays 
  of 
  a 
  thin 
  pencil 
  

   which 
  can 
  be 
  cut 
  at 
  right 
  angles 
  by 
  a 
  surface 
  pass 
  through 
  

   two 
  lines, 
  such 
  that 
  the 
  planes 
  containing 
  either 
  of 
  them 
  and 
  

   the 
  principal 
  ray 
  are 
  perpendicular 
  to 
  each 
  other. 
  

  

  7. 
  Again, 
  since 
  the 
  equation 
  of 
  a 
  surface 
  near 
  the 
  origin 
  

   with 
  the 
  axis 
  of 
  z 
  along 
  the 
  normal 
  at 
  the 
  origin 
  is, 
  to 
  the 
  

   second 
  order, 
  

  

  Pi 
  P2 
  

  

  the 
  characteristic 
  function 
  (V 
  = 
  constant) 
  for 
  a 
  thin 
  pencil 
  in 
  

   a 
  medium 
  //, 
  with 
  the 
  axial 
  ray 
  proceeding 
  along 
  the 
  axis 
  

   of 
  z 
  is 
  

  

  M-£-& 
  

  

  approximately, 
  if 
  the 
  aberration 
  is 
  neglected. 
  With 
  the 
  

   help 
  of 
  this 
  equation, 
  the 
  problem 
  of 
  reflexion 
  and 
  refraction 
  

   of 
  direct 
  and 
  oblique 
  pencils 
  (aberration 
  being 
  neglected) 
  

   can 
  be 
  treated 
  in 
  the 
  usual 
  way. 
  

  

  8. 
  To 
  illustrate 
  this, 
  consider 
  the 
  following 
  example 
  : 
  — 
  

   A 
  pencil 
  of 
  rays 
  is 
  refracted 
  through 
  a 
  prism 
  and 
  the 
  axis 
  

   of 
  the 
  pencil 
  is 
  constantly 
  in 
  a 
  principal 
  plane 
  of 
  the 
  prism 
  : 
  

   also 
  the 
  angular 
  position 
  of 
  the 
  focal 
  lines 
  at 
  incidence 
  and 
  

   after 
  the 
  first 
  and 
  second 
  refraction 
  relative 
  to 
  the 
  edge 
  of 
  

   the 
  prism 
  is 
  defined 
  by 
  a, 
  /5, 
  7, 
  and 
  the 
  distances 
  of 
  the 
  

   initial 
  and 
  final 
  foci 
  from 
  the 
  first 
  and 
  the 
  second 
  surface 
  

   are 
  u 
  l9 
  u 
  2 
  and 
  v 
  lt 
  v 
  2 
  respectively. 
  

  

  Let 
  us 
  also 
  take 
  Ui, 
  U 
  2 
  as 
  the 
  distances 
  of 
  the 
  foci 
  after 
  

   ■the 
  first 
  refraction, 
  from 
  the 
  first 
  surface, 
  and 
  V 
  1? 
  V 
  2 
  these 
  

   distances 
  from 
  the 
  second 
  surface. 
  

  

  