﻿148 
  Prof. 
  D. 
  N. 
  Mallik 
  on 
  Fermat's 
  Law. 
  

  

  Therefore, 
  in 
  

  

  x 
  2 
  y 
  2 
  

  

  2z=- 
  + 
  

   Pi 
  P2 
  

   we 
  must 
  write 
  

  

  V 
  V 
  % 
  ptr 
  

  

  instead 
  of 
  z, 
  in 
  order 
  to 
  be 
  correct 
  up 
  to 
  this 
  order. 
  

  

  Hence 
  the 
  equation 
  of 
  the 
  characteristic 
  surface 
  (V 
  = 
  con- 
  

   stant) 
  is 
  of 
  the 
  form 
  

  

  y 
  ■ 
  z 
  -l°L 
  + 
  it+Ji 
  X 
  l 
  + 
  lt\ 
  

  

  + 
  ax 
  % 
  + 
  ?>bx 
  2 
  y 
  + 
  Sexy 
  2 
  + 
  dy 
  % 
  (say), 
  

  

  the 
  terms 
  in 
  z 
  2 
  and 
  e 
  3 
  being 
  neglected, 
  as 
  being 
  of 
  higher 
  

   orders. 
  

  

  The 
  projection 
  of 
  this, 
  on 
  the 
  principal 
  plane 
  y 
  = 
  is 
  

  

  x 
  2 
  x 
  2 
  

  

  z—\ 
  Y\z—<, 
  + 
  ax 
  z 
  . 
  

  

  h 
  Pi 
  

  

  The 
  equation 
  of 
  the 
  normal 
  to 
  this, 
  at 
  x\ 
  z\ 
  is 
  

  

  x 
  — 
  x' 
  z 
  — 
  z' 
  

  

  Pi 
  Pi 
  2 
  Pi 
  2 
  

  

  This 
  meets 
  the 
  normal 
  through 
  the 
  origin 
  at 
  the 
  point 
  z, 
  0, 
  

  

  where 
  

  

  r' 
  2 
  

   1 
  ^ 
  i 
  

  

  272" 
  1 
  

  

  r 
  P^ 
  

  

  pi 
  Pi 
  

  

  Therefore,, 
  the 
  aberration 
  in 
  the 
  principal 
  plane 
  is 
  given 
  

  

  by 
  

  

  r 
  '2 
  

  

  1® 
  I 
  

  

  2 
  _ 
  2 
  X 
  

  

  Pi 
  Pi 
  2 
  

  

  r. 
  ^' 
  2 
  s' 
  -i 
  

  

  ^L*S 
  5 
  + 
  p 
  1 
  +3 
  ^J 
  

  

  Pi 
  Pi 
  2 
  

   = 
  — 
  3ax'p 
  lf 
  rejecting 
  terms 
  of 
  higher 
  

   x 
  i 
  [orders, 
  

  

  = 
  — 
  ^ap 
  x 
  2 
  x 
  , 
  where 
  — 
  = 
  $i. 
  

   Pi 
  

  

  