﻿Achromatic 
  Objectives 
  of 
  Telescopes. 
  289 
  

  

  From 
  the 
  figure 
  we 
  have 
  

  

  y 
  : 
  F 
  % 
  :: 
  BEtw 
  co 
  : 
  F 
  % 
  -BE 
  

   whence, 
  

  

  __M_ 
  (1) 
  

  

  2/ 
  + 
  i^ 
  3 
  tan 
  o> 
  w 
  

  

  and 
  again 
  from 
  the 
  figure 
  

  

  y 
  : 
  F 
  2 
  : 
  : 
  S^tan 
  o) 
  : 
  BH-F, 
  

   whence, 
  

  

  ^5T 
  = 
  ^ 
  (2) 
  

  

  ?/ 
  — 
  A 
  2 
  tan 
  <o 
  

  

  Then, 
  to 
  make 
  BE— 
  BH, 
  we 
  equate 
  the 
  right 
  hand 
  members 
  

   of 
  (1) 
  and 
  (2) 
  and 
  obtain 
  

  

  F 
  F 
  

  

  y— 
  i^tanco 
  y-fi^tanco 
  v 
  ; 
  

  

  which 
  is 
  the 
  condition 
  for 
  the 
  destruction 
  of 
  the 
  oblique 
  aber- 
  

   ration. 
  

  

  If 
  now 
  we 
  suppose 
  the 
  spherical 
  aberration 
  of 
  the 
  lens 
  AG 
  

   to 
  be 
  perfectly 
  corrected 
  for 
  all 
  objects 
  situated 
  in 
  its 
  axis, 
  

   whether 
  near 
  or 
  far, 
  and 
  if 
  F 
  2 
  and 
  F 
  3 
  are 
  the 
  focal 
  points 
  for 
  

   rays 
  diverging 
  respectively 
  from 
  the 
  radiants 
  J? 
  2 
  and 
  R 
  z 
  then 
  

   we 
  shall 
  have 
  rigorously, 
  for 
  all 
  points 
  of 
  the 
  lens 
  

  

  l/F=llF 
  % 
  + 
  \/B 
  % 
  

  

  l/F=l/F 
  z 
  + 
  l/B 
  3 
  

   From 
  the 
  figure 
  

  

  B 
  2 
  —B 
  1 
  : 
  i2 
  2 
  : 
  : 
  B 
  1 
  tan 
  (o 
  : 
  y 
  

  

  whence, 
  

  

  1 
  1 
  tan 
  w 
  

  

  W 
  

  

  (5) 
  

  

  ^ 
  A 
  y 
  

  

  and 
  again 
  from 
  the 
  figure 
  

  

  R 
  x 
  — 
  B 
  3 
  : 
  B 
  z 
  :: 
  B 
  l 
  tan 
  co 
  : 
  y 
  

  

  whence, 
  

  

  1 
  1 
  tan 
  co 
  , 
  . 
  

  

  Rr^ 
  + 
  ~ 
  (6) 
  

  

  By 
  substituting 
  the 
  right 
  hand 
  members 
  of 
  (5) 
  and 
  (6) 
  in 
  (4)j 
  

   we 
  obtain, 
  

  

  111 
  tan 
  <o 
  tan 
  co 
  

  

  whence, 
  

  

  tan 
  

  

  CO 
  

  

  i 
  

  

  y 
  

  

  K 
  

  

  

  K 
  

  

  

  y— 
  i^tanco 
  y 
  + 
  F 
  2 
  tan 
  co 
  

  

  