﻿290 
  HarTcness 
  — 
  Best 
  Form 
  for 
  the 
  Double 
  

  

  which 
  is 
  identical 
  with 
  (3) 
  ; 
  and 
  as 
  the 
  validity 
  of 
  (7) 
  depends 
  

   only 
  upon 
  the 
  exact 
  fulfillment 
  of 
  the 
  condition 
  expressed 
  by 
  

   the 
  equations 
  (4), 
  while 
  (3) 
  implies 
  the 
  destruction 
  of 
  the 
  

   oblique 
  aberration, 
  we 
  conclude 
  that 
  

  

  If 
  in 
  any 
  thin 
  lens 
  the 
  spherical 
  aberration 
  is 
  completely 
  

   corrected 
  for 
  all 
  objects 
  situated 
  in 
  the 
  axis 
  of 
  the 
  lens, 
  then 
  

   the 
  oblique 
  aberration 
  will 
  also 
  be 
  completely 
  corrected. 
  

  

  Thus 
  the 
  problem 
  of 
  obtaining 
  equally 
  distinct 
  images 
  in 
  all 
  

   parts 
  of 
  the 
  field 
  of 
  view 
  is 
  reduced 
  to 
  the 
  problem 
  of 
  making 
  

   the 
  destruction 
  of 
  the 
  spherical 
  aberration 
  for 
  axial 
  rays 
  inde- 
  

   pendent 
  of 
  the 
  distance 
  of 
  the 
  radiant 
  from 
  the 
  telescope, 
  and 
  

   we 
  have 
  now 
  to 
  inquire 
  how 
  far 
  that 
  can 
  be 
  done 
  in 
  the 
  case 
  

   of 
  double 
  achromatic 
  objectives. 
  Fortunately 
  the 
  latter 
  prob- 
  

   lem 
  is 
  not 
  new, 
  having 
  presented 
  itself 
  to 
  Sir 
  J. 
  F. 
  W. 
  Herschel 
  

   so 
  long 
  ago 
  as 
  1821, 
  when 
  he 
  showed 
  that 
  the 
  spherical 
  aberra- 
  

   tion 
  of 
  an 
  achromatic 
  objective 
  consisting 
  of 
  any 
  number 
  of 
  

   thin 
  lenses 
  in 
  contact 
  can 
  be 
  expressed 
  in 
  the 
  form 
  

  

  &f=if(X+YjDi-ZD>) 
  (8) 
  

  

  where 
  y 
  is 
  the 
  semi-aperture 
  of 
  the 
  objective, 
  X, 
  ]Tand 
  Z 
  

   are 
  functions 
  of 
  the 
  refractive 
  indexes 
  and 
  focal 
  distances 
  of 
  

   the 
  several 
  lenses, 
  and 
  of 
  the 
  radii 
  of 
  curvature 
  of 
  their 
  first 
  

   surfaces, 
  and 
  D 
  is 
  the 
  reciprocal 
  of 
  the 
  distance 
  of 
  the 
  object 
  

   viewed. 
  Of 
  course 
  the 
  refractive 
  indexes 
  are 
  known, 
  and 
  the 
  

   focal 
  length 
  of 
  the 
  objective 
  together 
  with 
  the 
  achromatic 
  

   equation 
  determine 
  the 
  focal 
  distances 
  of 
  the 
  several 
  lenses. 
  

   Consequently 
  equation 
  (8) 
  contains 
  as 
  indeterminates 
  only 
  the 
  

   radii 
  of 
  curvature 
  of 
  one 
  surface 
  of 
  each 
  lens; 
  and 
  in 
  the 
  case 
  

   of 
  a 
  double 
  objective, 
  two 
  independent 
  conditions 
  suffice 
  to 
  

   determine 
  them. 
  Herschel 
  concluded 
  that 
  these 
  conditions 
  

   should 
  be 
  

  

  X= 
  T= 
  (9) 
  

  

  because 
  the 
  term 
  depending 
  on 
  D 
  is 
  thus 
  destroyed 
  for 
  all 
  dis- 
  

   tances 
  of 
  the 
  radiant, 
  and 
  although 
  the 
  term 
  in 
  D 
  2 
  still 
  remains, 
  

   it 
  is 
  usually 
  small 
  enough 
  to 
  be 
  neglected 
  whenever 
  D 
  does 
  

   not 
  exceed 
  0*0 
  L 
  of 
  the 
  length 
  of 
  the 
  telescope. 
  He 
  summed 
  

   up 
  the 
  situation 
  by 
  saying,* 
  

  

  " 
  If 
  these 
  equations 
  be 
  combined, 
  we 
  shall 
  obtain 
  the 
  dimen- 
  

   sions 
  of 
  an 
  object-glass 
  free 
  from 
  aberration, 
  both 
  for 
  celestial 
  

   and 
  terrestrial 
  objects, 
  provided 
  we 
  restrict 
  our 
  views 
  to 
  objects 
  

   situated 
  in 
  the 
  prolongation 
  of 
  the 
  axis 
  of 
  the 
  telescope." 
  

  

  That 
  proviso 
  is 
  shown 
  by 
  the 
  present 
  paper 
  to 
  be 
  a 
  mistake, 
  

   but 
  from 
  Herschel's 
  point 
  of 
  view 
  it 
  seemed 
  necessary, 
  because 
  

   his 
  equations 
  contained 
  no 
  explicit 
  provision 
  for 
  the 
  destruc- 
  

   tion 
  of 
  the 
  lateral 
  aberration. 
  

  

  *Phil. 
  Trans., 
  1821, 
  p. 
  260. 
  

  

  