﻿of 
  correcting 
  Telescopic 
  Objectives. 
  475 
  

  

  S 
  

   and 
  the 
  comatic 
  error 
  is 
  —^f/i(^i 
  — 
  h) 
  2 
  (p 
  1 
  -\-p 
  2 
  ) 
  3 
  .~, 
  and 
  ex- 
  

   pression 
  (2) 
  is 
  (S 
  2 
  — 
  &i)iii 
  ; 
  but 
  if 
  the 
  object 
  is 
  at 
  infinity, 
  and 
  

   the 
  raj 
  from 
  it 
  strikes 
  at 
  height 
  h 
  and 
  slope 
  0, 
  the 
  spherical 
  

  

  S 
  

   aberration 
  is 
  then 
  {0 
  — 
  h(p 
  1 
  +p 
  2 
  )} 
  3 
  . 
  ~, 
  the 
  comatic 
  error 
  is 
  

  

  30{6-h(p 
  1+1 
  , 
  2 
  )y.^\ 
  and 
  expression 
  (2) 
  is 
  (S 
  l 
  -S 
  2 
  )(p 
  1 
  + 
  J 
  9 
  2 
  ); 
  

  

  and 
  in 
  either 
  case 
  expression 
  (1) 
  is 
  ^Sj. 
  Of 
  course 
  only 
  

   rays 
  in 
  a 
  meridian 
  plane 
  through 
  the 
  object 
  are 
  here 
  

   considered. 
  

  

  For 
  the 
  purpose 
  of 
  merely 
  calculating 
  the 
  aberrations, 
  

   without 
  discussion, 
  I 
  find 
  it 
  very 
  convenient 
  to 
  follow 
  

   Professor 
  Conrady's 
  plan 
  of 
  introducing 
  a 
  fictitious 
  air-gap 
  

   between 
  all 
  cemented 
  surfaces, 
  and 
  then 
  computing 
  (by 
  slide- 
  

   rule) 
  the 
  aberrations 
  for 
  each 
  lens, 
  rather 
  than 
  for 
  each 
  sur- 
  

   face. 
  The 
  ray 
  is 
  first 
  taken 
  through 
  the 
  system 
  by 
  the 
  schedule 
  

   ii 
  = 
  c 
  1 
  — 
  a 
  1 
  , 
  Nti'=ti, 
  Ui=ci 
  — 
  ii, 
  u 
  2 
  = 
  ui' 
  ; 
  and 
  so 
  on. 
  Then 
  

  

  the 
  form 
  taken 
  by 
  S 
  x 
  is 
  (l- 
  ^2R 
  2 
  (/i-"i') 
  + 
  (VJ^-V)}, 
  

  

  while 
  the 
  expression 
  / 
  1— 
  _W{hO'i 
  — 
  V) 
  -\-i 
  2 
  {u 
  2 
  — 
  i 
  2 
  ')} 
  is 
  

  

  S 
  2 
  — 
  Si 
  multiplied 
  by 
  u 
  l 
  or 
  by 
  — 
  P 
  as 
  the 
  case 
  may 
  be; 
  

   P 
  denotes 
  the 
  power 
  of 
  the 
  whole 
  combination, 
  pi+p 
  2 
  . 
  

  

  There 
  is 
  no 
  difficulty 
  about 
  modifying 
  the 
  coefficients 
  

   A, 
  B, 
  &c. 
  to 
  allow 
  for 
  a 
  third 
  component, 
  especially 
  as 
  most 
  

   examples 
  of 
  this 
  class 
  would 
  be 
  cemented., 
  so 
  that 
  all 
  terms 
  

   in 
  g 
  l 
  or 
  g 
  2 
  would 
  disappear. 
  

  

  I 
  now 
  proceed 
  to 
  deduce 
  some 
  consequences 
  of 
  the 
  above 
  

   formulae. 
  As 
  (2) 
  and 
  (1) 
  are 
  respectively 
  of 
  the 
  same 
  shape 
  

   as 
  the 
  equations 
  of 
  a 
  plane 
  and 
  a 
  conicoid, 
  it 
  is 
  natural 
  that 
  

   a 
  number 
  of 
  familiar 
  expressions 
  should 
  occur 
  in 
  the 
  

   deductions 
  from 
  them. 
  

  

  We 
  notice 
  first 
  that 
  (2) 
  is 
  linear. 
  Therefore 
  when 
  two 
  

   of 
  the 
  variables 
  (say 
  q 
  and 
  u 
  x 
  ) 
  have 
  been 
  fixed 
  to 
  remove 
  

   spherical 
  aberration, 
  there 
  is 
  only 
  one 
  value 
  of 
  the 
  third 
  which 
  

   will 
  remove 
  coma, 
  and 
  there 
  always 
  is 
  one; 
  i. 
  e., 
  with 
  the 
  

   form 
  of 
  the 
  leading 
  lens 
  fixed, 
  as 
  well 
  as 
  the 
  position 
  of 
  the 
  

   object, 
  there 
  is 
  only 
  one 
  air-gap, 
  and 
  therefore 
  only 
  one 
  form 
  

   appropriate 
  for 
  the 
  second 
  lens. 
  And 
  a 
  cemented 
  lens 
  cannot 
  

   be 
  free 
  from 
  coma 
  (except 
  by 
  good 
  luck) 
  if 
  c 
  x 
  and 
  u 
  x 
  have 
  

   been 
  fixed 
  by 
  other 
  considerations. 
  The 
  graph 
  of 
  the 
  comatic 
  

   error 
  is, 
  for 
  spherically 
  corrected 
  combinations, 
  a 
  straight 
  line, 
  

   whether 
  it 
  is 
  plotted 
  against 
  g, 
  or 
  u^ 
  or 
  any 
  of 
  the 
  c's. 
  For 
  

   all 
  other 
  combinations, 
  as 
  soon 
  as 
  two 
  variables 
  are 
  selected, 
  

  

  