﻿Focal 
  Plane 
  of 
  a 
  Telescope 
  with 
  Circular 
  Aperture, 
  5 
  

   or 
  since 
  2 
  J^ 
  (.«) 
  T 
  , 
  . 
  _ 
  , 
  s 
  

  

  the 
  above 
  equation 
  can 
  also 
  be 
  written 
  

  

  Jo(f) 
  =3 
  

   J 
  2 
  (#) 
  

  

  For 
  any 
  value 
  of 
  x, 
  Ji(#) 
  < 
  1 
  ; 
  the 
  relation 
  to 
  be 
  satisfied 
  

   at 
  the 
  inflexion-points, 
  

  

  Ji(#) 
  2x 
  

  

  shows 
  that 
  for 
  large 
  values 
  of 
  x, 
  J 
  (x) 
  must 
  be 
  very 
  small 
  ; 
  

   thus 
  the 
  abscissae 
  of 
  the 
  inflexion-points 
  belonging 
  to 
  this 
  set 
  

   will 
  ultimately 
  coincide 
  with 
  the 
  roots 
  of 
  

  

  joM=o. 
  

  

  If 
  in 
  the 
  relation 
  given 
  above 
  J 
  (#) 
  and 
  Ji(#) 
  be 
  expanded 
  

   in 
  semiconvergent 
  series, 
  we 
  get 
  the 
  equation 
  

  

  tanfa?- 
  x\=^^-0-5392. 
  - 
  +0-603. 
  -^ 
  - 
  ... 
  

  

  \ 
  4/ 
  11 
  x 
  x 
  4 
  

  

  The 
  ordinates 
  for 
  these 
  inflexion-points 
  are 
  given 
  by 
  

  

  9 
  

  

  j^Ji-m(i+£) 
  

  

  We 
  shall 
  distinguish 
  the 
  inflexion-points 
  given 
  by 
  the 
  roots 
  of 
  

  

  J 
  (x) 
  

   Ji(x) 
  =0 
  from 
  those 
  given 
  by 
  T 
  =3, 
  by 
  calling 
  the 
  former 
  

  

  inflexion-points 
  of 
  the 
  first 
  set 
  ; 
  and 
  the 
  latter 
  inflexion-points 
  

   of 
  the 
  second 
  set. 
  The 
  above 
  considerations 
  show 
  that 
  the 
  

   inflexion-points 
  of 
  the 
  second 
  set 
  lie 
  nearly 
  midway 
  between 
  

   those 
  of 
  the 
  first 
  set, 
  the 
  approximation 
  becoming 
  closer 
  

   for 
  increasing 
  values 
  of 
  x. 
  Further, 
  we 
  conclude 
  that 
  the 
  

   curve 
  has 
  neither 
  maximum 
  nor 
  minimum 
  (excepting 
  the 
  

   point 
  07 
  = 
  0, 
  j/ 
  = 
  l), 
  but 
  has 
  an 
  infinite 
  number 
  of 
  inflexion- 
  

   points 
  (of 
  the 
  first 
  set) 
  where 
  the 
  tangents 
  are 
  parallel 
  to 
  

   the 
  #-axis. 
  

  

  Between 
  the 
  inflexion-points 
  of 
  the 
  two 
  sets, 
  there 
  must 
  

   be 
  places 
  of 
  maximum 
  curvature 
  ; 
  these 
  points 
  are 
  given 
  by 
  

  

  J^) 
  _ 
  Z 
  X 
  \ 
  X 
  ) 
  - 
  5JoWJlW 
  - 
  2 
  -#^ 
  [3 
  + 
  2JA*) 
  + 
  4J 
  V) 
  J,«(*)] 
  

  

  A/ 
  X 
  

  

  16 
  Jo(aQJi 
  g 
  (*) 
  5i 
  J 
  iV) 
  =0 
  

  

  X* 
  X* 
  

  

  If 
  a? 
  be 
  large, 
  the 
  leading 
  term 
  is 
  evidently 
  Jo^^—Ji 
  2 
  ^), 
  

  

  