﻿oj 
  correcting 
  Telescopic 
  Objectives. 
  477 
  

  

  a 
  new 
  shape 
  (II). 
  It 
  is 
  easy 
  to 
  show 
  that 
  the 
  reciprocal 
  

   of 
  the 
  new 
  latus 
  rectum 
  is 
  A( 
  4^^ 
  ~"1 
  )• 
  At 
  the 
  vertex 
  of 
  

   this 
  parabola 
  we 
  not 
  only 
  satisfy 
  equation 
  (3), 
  but 
  also 
  

  

  Ec 
  1 
  + 
  Fw 
  1 
  + 
  2C^ 
  + 
  K=0 
  (4-) 
  

  

  We 
  therefore 
  find 
  the 
  vertex 
  by 
  solving 
  (3) 
  and 
  (4) 
  together 
  

   (ui 
  is 
  given), 
  which 
  gives 
  the 
  abscissa 
  c 
  1? 
  while 
  the 
  aberration 
  

   is 
  calculated 
  from 
  (1). 
  Having 
  the 
  vertex 
  and 
  the 
  latus 
  rectum 
  

   the 
  locus 
  is 
  quickly 
  drawn. 
  It 
  would 
  be 
  used 
  as 
  follows. 
  

   The 
  position 
  of 
  the 
  object 
  is 
  given 
  by 
  w 
  1? 
  and 
  we 
  wish 
  to 
  know 
  

   what 
  can 
  be 
  done 
  with 
  a 
  series 
  of 
  values 
  for 
  g. 
  Begin 
  with 
  one 
  

   of 
  them, 
  insert 
  it 
  and 
  u 
  x 
  in 
  (3), 
  and 
  find 
  c 
  v 
  Pick 
  out 
  the 
  point 
  

   on 
  parabola 
  II 
  which 
  has 
  this 
  Ci. 
  Lay 
  the 
  templet 
  for 
  

   parabola 
  I 
  with 
  its 
  vertex 
  at 
  this 
  point, 
  and 
  immediately 
  we 
  

   see 
  all 
  the 
  possible 
  aberrations 
  with 
  that 
  air-gap. 
  Now 
  

   change 
  g, 
  make 
  the 
  necessary 
  change 
  in 
  c 
  1 
  (by 
  simple 
  pro- 
  

   portion, 
  as 
  we 
  see 
  from 
  (3)), 
  shift 
  the 
  templet 
  to 
  this 
  new 
  

   vertex, 
  and 
  so 
  on. 
  

  

  Had 
  it 
  been 
  desired 
  to 
  vary 
  a^ 
  while 
  g 
  was 
  fixed, 
  the 
  only 
  

   difference 
  would 
  be 
  that 
  we 
  should 
  get 
  parabola 
  III 
  ; 
  its 
  

  

  iatus 
  rectum 
  is 
  the 
  reciprocal 
  of 
  A 
  (4^ 
  — 
  1 
  ) 
  and 
  its 
  vertex 
  

   is 
  given 
  by 
  using 
  \ 
  ' 
  

  

  DC 
  1 
  + 
  2Bm 
  1 
  + 
  F^ 
  + 
  H 
  = 
  0, 
  .... 
  (5) 
  

  

  instead 
  of 
  (4). 
  

  

  Finally, 
  suppose 
  we 
  wish 
  to 
  generalize 
  parabola 
  II 
  so 
  as 
  to 
  

   include 
  all 
  values 
  of 
  u 
  x 
  ; 
  we 
  must 
  first 
  find 
  the 
  locus 
  of 
  the 
  

   vertices 
  of 
  all 
  such 
  parabolas 
  as 
  II 
  while 
  u 
  x 
  varies. 
  It 
  will 
  

   again 
  be 
  a 
  parabola 
  (IY) 
  ; 
  its 
  latus 
  rectum 
  is 
  the 
  reciprocal 
  

  

  of 
  uquZef)* 
  ( 
  4ABC 
  + 
  DEF 
  - 
  AJF 
  2 
  -BE 
  2 
  - 
  CD 
  2 
  ), 
  and 
  its 
  

  

  vertex 
  is 
  found 
  by 
  solving 
  (3), 
  (4), 
  and 
  (5) 
  for 
  c 
  l5 
  and 
  then 
  

   finding 
  the 
  ordinate 
  from 
  (1). 
  

  

  If 
  III 
  is 
  generalized 
  in 
  this 
  way, 
  we 
  get 
  parabola 
  V, 
  with 
  

   the 
  same 
  vertex 
  as 
  IV, 
  but 
  its 
  latus 
  rectum 
  is 
  the 
  reciprocal 
  

  

  o£ 
  ^^jff^ 
  iABO 
  + 
  DEF-AF'-BE'-CD'). 
  

  

  To 
  use 
  V, 
  we 
  should 
  first 
  assign 
  a 
  value 
  to^; 
  solve 
  (3) 
  and 
  

   (5) 
  for 
  Ci, 
  and 
  pick 
  out 
  the 
  corresponding 
  point 
  on 
  V; 
  fit 
  

   to 
  that 
  point 
  as 
  vertex 
  the 
  templet 
  of 
  III, 
  and 
  draw 
  III. 
  

   Assign 
  a 
  value 
  to 
  u 
  u 
  solve 
  (3) 
  for 
  c 
  u 
  pick 
  out 
  the 
  point 
  on 
  III, 
  

   and 
  use 
  it 
  as 
  a 
  vertex 
  for 
  the 
  templet 
  of 
  I. 
  All 
  variations 
  in 
  

  

  Phil. 
  Mag. 
  S. 
  6. 
  Vol. 
  35. 
  No. 
  210. 
  June 
  1918. 
  2 
  L 
  

  

  