﻿4:76 
  Mr. 
  A. 
  0. 
  Allen 
  on 
  Grapliicat 
  Methods 
  

  

  the 
  graph 
  of 
  the 
  comatic 
  error 
  against 
  the 
  third 
  is 
  a 
  

   parabola. 
  

  

  Expression 
  (L) 
  is 
  quadratic 
  in 
  c 
  l} 
  u 
  u 
  or 
  g 
  ; 
  when 
  any 
  two 
  

   of 
  the 
  variables 
  have 
  been 
  fixed, 
  tliere 
  are 
  two 
  values 
  of 
  the 
  

   third 
  which 
  will 
  remove 
  spherical 
  aberration 
  (though 
  these 
  

   two 
  may 
  coincide, 
  or 
  be 
  imaginary). 
  With 
  g 
  and 
  u 
  x 
  fixed, 
  

   the 
  graph 
  of 
  the 
  aberration 
  against 
  any 
  one 
  of 
  the 
  c's 
  is 
  a 
  

   parabola 
  with 
  its 
  axis 
  vertical 
  ; 
  its 
  vertex 
  will 
  be 
  downward 
  

   if 
  pi 
  and 
  p 
  2 
  are 
  both 
  positive, 
  and 
  also 
  if 
  p, 
  is 
  positive 
  and 
  p 
  2 
  

   negative, 
  provided 
  the 
  combination 
  is 
  a 
  converging 
  one. 
  

   The 
  matter 
  upon 
  which 
  above 
  all 
  I 
  would 
  lay 
  stress 
  is 
  that 
  

   the 
  latus 
  rectum 
  of 
  this 
  parabola, 
  being 
  the 
  reciprocal 
  of 
  A, 
  

   is 
  entirely 
  independent 
  of 
  c 
  x 
  , 
  w 
  p 
  and 
  g 
  ; 
  in 
  other 
  words, 
  so 
  

   long 
  as 
  we 
  keep 
  to 
  the 
  same 
  two 
  glasses, 
  and 
  to 
  the 
  same 
  two 
  

   powers, 
  we 
  may 
  vary 
  the 
  curves 
  and 
  the 
  air-gap 
  and 
  the 
  

   position 
  of 
  the 
  object 
  as 
  much 
  as 
  we 
  please, 
  but 
  we 
  shall 
  

   always 
  get 
  the 
  same 
  parabola; 
  all 
  that 
  will 
  change 
  will 
  be 
  

   the 
  position 
  of 
  its 
  vertex. 
  Therefore 
  we 
  need 
  only 
  calculate 
  

   the 
  latus 
  rectum 
  (1/A), 
  plot 
  the 
  parabola 
  once 
  for 
  all 
  (either 
  

   by 
  the 
  focus-directrix 
  property 
  with 
  compnsses, 
  or 
  from 
  the 
  

   equation?/ 
  2 
  = 
  cx 
  with 
  logarithms 
  or 
  a 
  slide-rule), 
  and 
  then 
  cut 
  

   out 
  a 
  templet 
  of 
  this 
  parabolic 
  form. 
  Thereafter, 
  in 
  any 
  

   problem 
  on 
  lenses 
  of 
  these 
  two 
  glasses, 
  and 
  of 
  powers 
  pro- 
  

   portionate 
  to 
  pi 
  and 
  p 
  2 
  all 
  we 
  need 
  do 
  is 
  to 
  calculate 
  the 
  

   position 
  of 
  the 
  vertex, 
  lay 
  the 
  templet 
  in 
  position 
  and 
  draw 
  

   the 
  parabola 
  (I), 
  and 
  immediately 
  we 
  see 
  all 
  the 
  possibilities 
  

   within 
  reach 
  by 
  variations 
  of 
  the 
  four 
  curves. 
  We 
  can 
  see 
  

   what 
  curves 
  (if 
  any) 
  will 
  remove 
  spherical 
  error 
  (or 
  introduce 
  

   a 
  desired 
  amount), 
  and 
  whether 
  these 
  curves 
  will 
  also 
  correct 
  

   for 
  coma, 
  and 
  if 
  not, 
  how 
  bad 
  the 
  coma 
  will 
  be. 
  The 
  vertex 
  

   is 
  to 
  be 
  found 
  thus 
  : 
  — 
  Knowing 
  w 
  x 
  and 
  g, 
  calculate 
  c 
  x 
  from 
  

   the 
  equation 
  

  

  2Ac 
  1 
  + 
  D« 
  1 
  - 
  r 
  E(7 
  + 
  G 
  = 
  (3) 
  

  

  This 
  gives 
  the 
  abscissa 
  ; 
  then 
  calculate 
  the 
  ordinate 
  (the 
  

   aberration) 
  by 
  substituting 
  for 
  ci, 
  u 
  Xi 
  and 
  g 
  in 
  (1). 
  

  

  But 
  it 
  may 
  be 
  asked, 
  how 
  can 
  the 
  method 
  be 
  made 
  to 
  

   include 
  all 
  values 
  of 
  u^ 
  and 
  g, 
  when 
  the 
  above 
  graph 
  is 
  for 
  a 
  

   particular 
  u 
  Y 
  and 
  g 
  ? 
  Suppose 
  in 
  the 
  first 
  place 
  that 
  we 
  do 
  

   not 
  object 
  to 
  having 
  a 
  fixed 
  value 
  for 
  u 
  u 
  but 
  wish 
  to 
  exhibit 
  

   all 
  the 
  possibilities 
  which 
  follow 
  from 
  varying 
  g 
  as 
  well 
  as 
  c 
  x 
  . 
  

   We 
  begin 
  by 
  finding 
  the 
  locus 
  of 
  the 
  vertices 
  of 
  all 
  such 
  

   parabolas 
  as 
  the 
  above, 
  as 
  g 
  is 
  varied 
  with 
  u 
  x 
  fixed. 
  It 
  is 
  

   easy 
  to 
  see 
  what 
  this 
  locus 
  will 
  be 
  ; 
  it 
  means 
  plotting 
  the 
  

   aberration 
  (1) 
  with 
  the 
  restriction 
  (3) 
  imposed 
  upon 
  it 
  ; 
  and 
  

   as 
  (3) 
  is 
  linear, 
  the 
  graph 
  will 
  still 
  be 
  a 
  parabola, 
  although 
  of 
  

  

  