﻿of 
  correcting 
  Telescopic 
  Objectives. 
  473 
  

  

  C3 
  above 
  c 
  2 
  , 
  called 
  g, 
  is 
  the 
  air-gap 
  between 
  the 
  two 
  lenses 
  ; 
  

  

  , 
  ., 
  . 
  r 
  positive 
  1 
  ., 
  . 
  f 
  marginal 
  ) 
  m1 
  

  

  when 
  it 
  is 
  \ 
  L 
  . 
  \ 
  the 
  gap 
  is 
  1 
  . 
  , 
  \ 
  . 
  Ihe 
  

  

  i 
  ..v 
  tne 
  gap 
  is 
  i 
  . 
  , 
  v 
  

  

  [ 
  negative 
  J 
  ° 
  x 
  t 
  axial 
  J 
  

  

  refractive 
  index 
  of 
  the 
  first 
  lens 
  is 
  N 
  ; 
  of 
  the 
  second 
  n 
  ; 
  

   these 
  may 
  be 
  taken 
  to 
  refer, 
  for 
  instance, 
  to 
  the 
  brightest 
  

   part 
  of 
  the 
  spectrum. 
  Clearly 
  

  

  P\ 
  P\ 
  . 
  Pi 
  Pz 
  

  

  so 
  that 
  any 
  one 
  of 
  the 
  c's 
  determines 
  the 
  rest; 
  c 
  1 
  will 
  be 
  

   used 
  as 
  the 
  independent 
  one. 
  

  

  The 
  problem 
  is 
  to 
  express 
  (1) 
  the 
  spherical 
  aberration, 
  

   (2) 
  the 
  offence 
  against 
  the 
  sine-condition, 
  as 
  functions 
  of 
  

   N, 
  n, 
  c 
  1} 
  u 
  u 
  g. 
  The 
  only 
  rational 
  way 
  to 
  express 
  the 
  first 
  is 
  

   by 
  means 
  of 
  the 
  angle 
  between 
  the 
  emergent 
  ray 
  (in 
  air) 
  

   and 
  the 
  line 
  joining 
  the 
  point 
  of 
  emergence 
  to 
  the 
  ideal 
  

   image-point. 
  The 
  other 
  aberration 
  can 
  also 
  be 
  expressed 
  as 
  

   an 
  angle. 
  If 
  the 
  sine-condition 
  were 
  fulfilled 
  the 
  locus 
  of 
  

   the 
  intersections 
  of 
  corresponding 
  incident 
  and 
  emergent 
  

   rays 
  would 
  be 
  a 
  circle 
  of 
  curvature 
  Ui 
  + 
  uJ 
  (see 
  Steinheil 
  & 
  

   Voit, 
  p. 
  66 
  ; 
  Southall, 
  p. 
  409 
  ; 
  Cheshire, 
  J. 
  R. 
  Micr. 
  Soc). 
  

   The 
  excess 
  of 
  the 
  actual 
  curvature 
  above 
  u^ 
  -f 
  u± 
  is 
  the 
  angle 
  

   required. 
  The 
  expressions 
  I 
  obtain 
  are 
  these 
  : 
  — 
  

  

  (1) 
  The 
  spherical 
  aberration 
  is 
  

  

  Ac 
  x 
  2 
  + 
  B^ 
  2 
  + 
  Cg 
  2 
  + 
  Dciwi 
  + 
  Ec 
  lt 
  ? 
  + 
  Futf 
  + 
  Q 
  Cl 
  + 
  H^ 
  + 
  Kg 
  + 
  L 
  ; 
  

  

  . 
  . 
  '. 
  (1) 
  

  

  (2) 
  The 
  sine-error 
  is 
  

  

  Pc-i 
  + 
  Qwi 
  + 
  Ity 
  + 
  S, 
  (2) 
  

  

  where 
  

  

  A-ift(l-h 
  sr) 
  + 
  ift(l+ 
  1} 
  B=^ 
  1 
  (3+ 
  J) 
  + 
  j 
  ft 
  ( 
  8 
  + 
  1)., 
  

   ■-j^l+jL), 
  F=-^ 
  2 
  (2 
  + 
  ^); 
  

  

  