﻿478 
  Mr. 
  A. 
  0. 
  Allen 
  on 
  Graphical 
  Methods 
  

  

  aberration 
  as 
  the 
  curvatures 
  are 
  altered 
  are 
  now 
  shown 
  for 
  

   the 
  selected 
  g 
  and 
  u^ 
  

  

  I 
  propose 
  next 
  to 
  illustrate 
  the 
  use 
  of 
  these 
  methods 
  by- 
  

   solving 
  a 
  few 
  numerical 
  problems, 
  but 
  instead 
  of 
  employing 
  

   graphs 
  (which 
  would 
  involve 
  expensive 
  plates) 
  the 
  solutions 
  

   will 
  be 
  entirely 
  algebraical. 
  In 
  a 
  discussion 
  of 
  principles 
  it 
  

   it 
  desirable 
  to 
  use 
  5-figure 
  accuracy, 
  although 
  in 
  practice 
  

   4 
  figures 
  would 
  be 
  ample. 
  I 
  take 
  the 
  case 
  of 
  two 
  glasses 
  

   for 
  which 
  l/N 
  = 
  0-658805, 
  l/n 
  = 
  "61721; 
  also 
  ^ 
  = 
  2*508, 
  

   p 
  2 
  — 
  —1*508. 
  These 
  two 
  powers 
  were, 
  as 
  a 
  matter 
  of 
  fact, 
  

   selected 
  for 
  G 
  — 
  F 
  achromatism, 
  but 
  we 
  have 
  no 
  concern 
  

   with 
  that. 
  The 
  combination 
  is 
  one 
  which 
  was 
  studied 
  

   in 
  some 
  detail 
  in 
  Professor 
  Conrady's 
  autumn 
  class 
  last 
  

   year. 
  I 
  take 
  first 
  a 
  few 
  simple 
  questions 
  on 
  the 
  front 
  

   lens 
  alone 
  ; 
  p 
  2 
  must 
  then 
  be 
  put 
  equal 
  to 
  zero 
  in 
  the 
  coeffi- 
  

   cients; 
  A 
  -2-9063, 
  13 
  = 
  5-4143, 
  D= 
  -8*3206, 
  G= 
  -24*5088, 
  

   H 
  = 
  33*7267,L 
  = 
  67*7599. 
  

  

  Problem 
  (1). 
  If 
  the 
  object 
  is 
  at 
  infinity, 
  what 
  curves 
  give 
  

   least 
  spherical 
  aberration 
  ? 
  Solve 
  equation 
  (3) 
  with 
  w 
  x 
  and 
  

   ^ 
  = 
  0; 
  c 
  l 
  =— 
  Gr/2A 
  = 
  4*2165; 
  c 
  2 
  is 
  less 
  than 
  c 
  x 
  by 
  4*8426 
  in 
  

   each 
  example, 
  and 
  is 
  here 
  —0*6261; 
  the 
  so-called 
  "crossed 
  

   lens." 
  The 
  aberration, 
  according 
  to 
  (1), 
  is 
  16*089. 
  

  

  Problem 
  (2). 
  If 
  the 
  lens 
  is 
  given, 
  for 
  what 
  point 
  will 
  

   its 
  aberration 
  be 
  least? 
  For 
  example, 
  suppose 
  c'! 
  = 
  3*0, 
  

   c 
  2 
  = 
  -1*8426. 
  Solve 
  (5) 
  with 
  c 
  1 
  = 
  3 
  and 
  g 
  = 
  0; 
  w 
  1= 
  -(3D 
  

   -rH)/2B= 
  -0*80942, 
  i. 
  e. 
  the 
  object 
  is 
  about 
  12 
  focal 
  

   lengths 
  in 
  front 
  of 
  the 
  lens, 
  and 
  the 
  aberration 
  is 
  16*843. 
  

  

  Problem 
  (3). 
  Is 
  it 
  possible 
  for 
  a 
  single 
  thin 
  lens 
  to 
  be 
  

   free 
  from 
  aberration 
  ? 
  For 
  correction, 
  parabola 
  I 
  must 
  cross 
  

   the 
  horizontal 
  axis, 
  so 
  that 
  its 
  vertex 
  must 
  not 
  be 
  above 
  that 
  

   line. 
  The 
  critical 
  positions 
  are 
  where 
  parabola 
  III 
  crosses 
  

   the 
  line. 
  Its 
  vertex 
  is 
  given 
  by 
  solving 
  (3) 
  and 
  (5), 
  with 
  

  

  ,9 
  = 
  0; 
  these 
  give 
  u,= 
  -£=-1-254, 
  ^=2(fe) 
  = 
  24213 
  > 
  

  

  as 
  common-sense 
  shows. 
  For 
  these 
  values 
  the 
  aberration 
  

   is 
  16*9400. 
  Parabola 
  III 
  is 
  therefore 
  given 
  by: 
  "excess 
  

   aberration 
  above 
  16*9400 
  = 
  square 
  of 
  excess 
  curvature 
  above 
  

   2*4213." 
  Where 
  it 
  crosses 
  the 
  line, 
  aberration 
  =0, 
  there- 
  

   fore 
  <?! 
  = 
  10*4309 
  or 
  -5*5883, 
  and, 
  by 
  (3), 
  w, 
  =4*3396 
  or 
  

   — 
  6*8494. 
  So 
  that 
  aberration 
  cannot 
  be 
  corrected 
  except 
  

   for 
  object-points 
  within 
  about 
  \ 
  of 
  the 
  focal 
  length 
  on 
  one 
  

   side 
  (real) 
  and 
  \ 
  on 
  the 
  other 
  (virtual), 
  and 
  even 
  then 
  the 
  

   curves 
  have 
  to 
  be 
  very 
  strong. 
  There 
  is 
  no 
  useful 
  case. 
  As 
  

   a 
  mere 
  arithmetical 
  exercise, 
  we 
  may 
  take 
  u 
  x 
  — 
  —8; 
  then 
  q 
  

   has 
  to 
  be 
  -8*5342, 
  and 
  c 
  2 
  is 
  -13*377. 
  

  

  