﻿480 
  Graphical 
  Methods 
  of 
  correcting 
  Telescopic 
  Objectives. 
  

  

  It 
  crosses 
  the 
  axis 
  where 
  -0*6965= 
  -*103042(c 
  1 
  -0*24933) 
  2 
  , 
  

   i. 
  e. 
  where 
  c 
  t 
  = 
  2*8492 
  and 
  -2*3505, 
  and 
  (3) 
  gives 
  for 
  the 
  

   corresponding 
  values 
  i^ 
  L 
  = 
  0*8398 
  to 
  — 
  2*8497. 
  Provided 
  

   that 
  u 
  1 
  does 
  not 
  lie 
  within 
  this 
  range, 
  an 
  air-gap 
  1 
  is 
  not 
  

   incompatible 
  with 
  spherical 
  correction. 
  

  

  Problem 
  (10). 
  What 
  is 
  the 
  largest 
  air-gap 
  which 
  could 
  be 
  

   substituted 
  for 
  0*1 
  in 
  problem 
  (8) 
  so 
  as 
  to 
  make 
  spherical 
  

   correction 
  possible 
  ? 
  This 
  time 
  parabola 
  II 
  is 
  needed. 
  For 
  

   the 
  vertex, 
  solve 
  (3) 
  and 
  (4) 
  with 
  i^ 
  = 
  0; 
  namely, 
  c 
  1 
  = 
  4*2164. 
  

   aberration 
  =14*201. 
  It 
  crosses 
  the 
  axis 
  where 
  —14*201 
  = 
  

   -2*1072(c 
  1 
  -4*2164) 
  2 
  , 
  i. 
  e. 
  where 
  c 
  l= 
  = 
  1-6204 
  or 
  6*Sl24, 
  and 
  

   (3) 
  then 
  gives 
  for 
  g 
  the 
  values 
  0-06726 
  and 
  3*8317. 
  So 
  that 
  

   for 
  spherical 
  correction 
  for 
  a 
  distant 
  object 
  the 
  air-gap 
  must 
  

   not 
  exceed 
  0*06726 
  (unless 
  it 
  has 
  an 
  absurdly 
  high 
  value). 
  

  

  Problem 
  (11). 
  If 
  ii\ 
  lies 
  outside 
  a 
  certain 
  range, 
  all 
  values 
  

   of 
  the 
  air-gap 
  are 
  compatible 
  with 
  spherical 
  correction 
  ; 
  what 
  

   is 
  this 
  range? 
  Parabola 
  IV" 
  supplies 
  the 
  answer. 
  Its 
  vertex 
  

   is 
  found 
  by 
  solving 
  (3), 
  (4\ 
  and 
  (5) 
  simultaneously; 
  namely, 
  

   Cl 
  = 
  3*6073, 
  ^=-0*42546, 
  and 
  # 
  = 
  19426; 
  aberration 
  = 
  

   14-2374. 
  Its 
  equation 
  is: 
  (aberration 
  -14*2374) 
  = 
  - 
  0*099481 
  

   (0,-3 
  6073) 
  2 
  . 
  It 
  crosses 
  the 
  axis 
  at 
  ^ 
  = 
  15*570 
  or 
  -8*356, 
  

   which 
  means, 
  by 
  equations 
  (3) 
  and 
  (4), 
  that 
  u 
  1 
  = 
  7*931 
  or 
  

  

  — 
  8*783. 
  For 
  stronger 
  convergence 
  or 
  divergence 
  than 
  

   this, 
  correction 
  is 
  possible 
  with 
  any 
  air-gap, 
  provided 
  the 
  

   curvatures 
  are 
  chosen 
  properly. 
  

  

  Problem 
  (12) 
  . 
  Returning 
  to 
  problem 
  (7), 
  let 
  us 
  find 
  the 
  

   comatic 
  error 
  for 
  the 
  two 
  spherically 
  corrected 
  lenses 
  there 
  

   given. 
  Expression 
  (2) 
  gives 
  (for 
  q 
  = 
  0*60806) 
  the 
  value 
  

  

  — 
  1*62092; 
  and 
  for 
  ^ 
  = 
  2*4473, 
  the 
  error 
  is 
  1*5454. 
  Equating 
  

   (2) 
  to 
  zero, 
  with 
  u 
  x 
  =0 
  and# 
  = 
  0,we 
  have 
  c 
  x 
  — 
  1*5496 
  to 
  comply 
  

   with 
  the 
  sine-condition. 
  What 
  the 
  manufacturer 
  would 
  pro- 
  

   bably 
  do, 
  if 
  he 
  were 
  tied 
  down 
  to 
  these 
  two 
  glasses 
  and 
  a 
  

   cemented 
  doublet, 
  would 
  be 
  to 
  adopt 
  a 
  compromise; 
  and 
  this 
  

   again 
  shows 
  the 
  difficulty 
  of 
  dealing 
  with 
  lenses 
  by 
  tables, 
  

   for 
  no 
  tables 
  can 
  be 
  so 
  voluminous 
  as 
  to 
  provide 
  for 
  

   compromises. 
  

  

  Problem 
  (13). 
  Giving 
  up 
  the 
  idea 
  of 
  a 
  cemented 
  doublet, 
  

   let 
  us 
  find 
  what 
  air-gap 
  will 
  remove 
  both 
  spherical 
  aberration 
  

   and 
  coma 
  for 
  a 
  distant 
  object. 
  Equate 
  (1) 
  and 
  (2) 
  to 
  zero, 
  and 
  

   solve 
  with 
  1^ 
  = 
  0. 
  The 
  result 
  is:y='06721 
  (or 
  3'8339, 
  which 
  

   does 
  not 
  matter), 
  c 
  1 
  = 
  1*6448, 
  c 
  2 
  =— 
  3*1978, 
  c 
  3 
  =— 
  3*1306, 
  

   c 
  4 
  = 
  —0*6991. 
  These 
  curves, 
  then, 
  form 
  a 
  favourable 
  starting 
  

   point 
  for 
  calculating 
  an 
  achromatic 
  aplanat. 
  They 
  would 
  be 
  

   improved 
  by 
  making 
  small 
  allowances 
  for 
  the 
  thickness 
  of 
  

   the 
  lenses, 
  but 
  this 
  matter 
  must 
  be 
  postponed. 
  It 
  also 
  

   remains 
  to 
  show 
  what 
  actual 
  residue 
  of 
  error 
  remains 
  in 
  all 
  

  

  