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  Harhiess 
  — 
  Best 
  Form 
  for 
  the 
  Double 
  

  

  The 
  conditions 
  which 
  should 
  be 
  satisfied 
  by 
  a 
  perfect 
  objec- 
  

   tive 
  are, 
  first, 
  that 
  it 
  shall 
  have 
  a 
  specified 
  focal 
  length 
  ; 
  

   second, 
  that 
  its 
  achromatic 
  aberration 
  shall 
  be 
  corrected, 
  which, 
  

   strictly 
  speaking, 
  would 
  include 
  the 
  destruction 
  of 
  the 
  second- 
  

   ary 
  spectrum, 
  but 
  that 
  point 
  will 
  not 
  be 
  considered 
  in 
  the 
  

   present 
  paper 
  ; 
  and 
  third, 
  that 
  its 
  spherical 
  aberration 
  shall 
  be 
  

   corrected, 
  not 
  only 
  for 
  distant 
  objects 
  lying 
  in 
  its 
  optical 
  axis, 
  

   but 
  for 
  all 
  objects, 
  whether 
  near 
  or 
  remote, 
  situated 
  anywhere 
  

   within 
  its 
  field 
  of 
  view. 
  It 
  has 
  been 
  surmised 
  that 
  the 
  latter 
  

   condition 
  was 
  what 
  Fraunhofer 
  aimed 
  at 
  in 
  the 
  object-glasses 
  

   which 
  he 
  constructed, 
  but 
  I 
  am 
  not 
  aware 
  that 
  the 
  mathemat- 
  

   ical 
  theory 
  involved 
  in 
  the 
  said 
  condition 
  has 
  ever 
  been 
  formu- 
  

   lated, 
  and, 
  therefore, 
  it 
  seems 
  worth 
  while 
  to 
  examine 
  it 
  here. 
  

  

  Referring 
  to 
  the 
  figure, 
  let 
  AC 
  be 
  a 
  thin 
  lens 
  whose 
  optical 
  

   axis 
  is 
  RJd, 
  and 
  let 
  F 
  2 
  and 
  F 
  3 
  be, 
  respectively, 
  the 
  focal 
  points 
  

   for 
  axial 
  rays 
  proceeding 
  from 
  the 
  radiants 
  R 
  2 
  and 
  R 
  3 
  . 
  Further- 
  

   more, 
  let 
  R 
  1 
  be 
  an 
  oblique 
  radiant 
  situated 
  on 
  the 
  straight 
  line 
  

   joining 
  R 
  2 
  with 
  A, 
  and 
  let 
  the 
  rays 
  R 
  X 
  AI 
  and 
  RfiG 
  proceed 
  

   from 
  it. 
  The 
  straight 
  line 
  R^BGI, 
  passing 
  through 
  the 
  opti- 
  

   cal 
  center 
  of 
  the 
  lens, 
  will 
  be 
  intersected 
  by 
  them 
  at 
  G 
  and 
  I, 
  

   and 
  consequently 
  images 
  of 
  R 
  x 
  will 
  be 
  formed 
  throughout 
  the 
  

   entire 
  interval 
  between 
  these 
  points. 
  Finally, 
  draw 
  the 
  lines 
  

   R 
  X 
  D, 
  EG 
  and 
  HI 
  perpendicular 
  to 
  the 
  optical 
  axis 
  R 
  2 
  H. 
  

  

  For 
  brevity 
  let 
  

  

  F 
  = 
  Principal 
  focal 
  distance 
  of 
  the 
  lens 
  AC. 
  

   y 
  = 
  AB 
  = 
  AC 
  = 
  semi-diameter 
  of 
  the 
  lens 
  A 
  C. 
  

   B 
  1 
  = 
  Distance 
  DB. 
  JF[ 
  = 
  Distance 
  F^B. 
  

   B 
  n 
  — 
  Distance 
  BB. 
  F, 
  — 
  Distance 
  FB. 
  

  

  B„ 
  = 
  Distance 
  BB. 
  

  

  = 
  The 
  angle 
  B 
  X 
  BD 
  = 
  IBB. 
  

  

  An 
  examination 
  of 
  the 
  figure 
  shows 
  that 
  the 
  oblique 
  spher- 
  

   ical 
  aberration 
  arising 
  from 
  the 
  eccentric 
  position 
  of 
  the 
  radiant 
  

   R 
  x 
  is 
  represented 
  by 
  the 
  interval 
  EH 
  between 
  the 
  images 
  at 
  G 
  

   and 
  I, 
  and 
  in 
  order 
  to 
  destroy 
  it, 
  the 
  point 
  G 
  must 
  be 
  made 
  to 
  

   coincide 
  with 
  the 
  point 
  I. 
  That 
  will 
  be 
  the 
  case 
  when 
  BE 
  — 
  

   BH, 
  and 
  we 
  have 
  now 
  to 
  inquire 
  what 
  conditions 
  must 
  subsist 
  

   in 
  order 
  to 
  bring!about 
  this 
  equality. 
  

  

  