﻿472 
  Mr. 
  A. 
  0. 
  Allen 
  on 
  Graphical 
  Methods 
  

  

  power. 
  All 
  the 
  lenses 
  are 
  achromatic 
  doublets 
  for 
  C 
  and 
  F, 
  

   whereas 
  in 
  practice 
  the 
  objective 
  will 
  often 
  be 
  required 
  to 
  

   have 
  some 
  chromatic 
  error. 
  All 
  the 
  lenses 
  are 
  cemented, 
  so 
  

   that 
  they 
  are 
  corrected 
  either 
  for 
  sphericity 
  or 
  for 
  coma, 
  but 
  

   not 
  for 
  both 
  ; 
  no 
  tables 
  of 
  moderate 
  bulk 
  could 
  include 
  the 
  

   possibility 
  of 
  air-gaps. 
  

  

  Again, 
  all 
  the 
  lenses 
  are 
  computed 
  for 
  an 
  object 
  at 
  infinity; 
  

   whereas 
  reading-telescopes 
  should 
  be 
  computed 
  for 
  a 
  com- 
  

   paratively 
  near 
  object. 
  Finally, 
  the 
  tables 
  refer 
  only 
  to 
  

   doublets, 
  whereas 
  the 
  methods 
  given 
  below 
  can 
  be 
  applied 
  

   also 
  to 
  triplets; 
  or, 
  for 
  that 
  matter, 
  to 
  systems 
  with 
  any 
  

   number 
  of 
  thin 
  components, 
  but 
  combinations 
  of 
  four 
  or 
  five 
  

   lenses 
  as 
  telescopic 
  objectives 
  are 
  to 
  be 
  regarded 
  as 
  mere 
  

   scientific 
  bizarreries. 
  Now 
  all 
  these 
  variables 
  (refractive 
  

   index, 
  dispersive 
  power, 
  position 
  of 
  object, 
  air-gaps, 
  number 
  

   of 
  components) 
  are 
  taken 
  account 
  of 
  below, 
  and 
  without 
  any 
  

   serious 
  addition 
  to 
  the 
  labour 
  involved. 
  But 
  it 
  must 
  always 
  

   be 
  remembered 
  that 
  the 
  results 
  arrived 
  at 
  in 
  this 
  way 
  (or 
  by 
  

   the 
  tables) 
  ought 
  never 
  to 
  be 
  seen 
  by 
  the 
  lens-grinder; 
  they 
  

   are 
  simply 
  intended 
  to 
  give 
  tbe 
  computer 
  a 
  favourable 
  start. 
  

  

  The 
  assumptions 
  made 
  are: 
  (1) 
  that 
  the 
  thickness 
  of 
  each 
  

   lens 
  or 
  gap 
  is 
  negligible 
  ; 
  (2) 
  that 
  all 
  the 
  angles 
  in 
  the 
  calcu- 
  

   lation 
  are 
  so 
  small 
  that 
  the 
  excess 
  of 
  any 
  angle 
  above 
  its 
  

   sine 
  is 
  exactly 
  equal 
  to 
  a 
  sixth 
  of 
  the 
  cube 
  of 
  the 
  angle. 
  In 
  

   other 
  words, 
  the 
  rays 
  could 
  all 
  travel 
  within 
  a 
  capillary 
  tube 
  

   lying 
  along 
  the 
  axis 
  of 
  the 
  lens. 
  

  

  The 
  symbols 
  employed 
  below 
  are 
  chosen 
  to 
  suit 
  the 
  present 
  

   problem, 
  and 
  would 
  not 
  necessarily 
  fit 
  into 
  a 
  more 
  general 
  

   scheme. 
  Focal 
  lengths, 
  radii, 
  and 
  intersection-distances 
  are 
  

   avoided; 
  it 
  is 
  the 
  reciprocals 
  of 
  these 
  quantities 
  which 
  are 
  

   more 
  important. 
  The 
  four 
  curvatures 
  are 
  c 
  1? 
  c 
  2 
  , 
  c 
  3 
  , 
  c 
  4 
  , 
  from 
  

   left 
  to 
  right; 
  the 
  powers 
  of 
  the 
  two 
  lenses 
  are 
  p 
  u 
  p 
  2 
  , 
  and 
  it 
  

   is 
  assumed 
  that 
  the 
  power 
  of 
  the 
  combination 
  is 
  chosen 
  as 
  a 
  

   unit, 
  so 
  that 
  p 
  l 
  + 
  p 
  2 
  = 
  l. 
  So 
  far 
  as 
  this 
  paper 
  is 
  concerned 
  

   there 
  is 
  no 
  condition 
  whatever 
  connecting 
  p 
  l 
  and 
  p 
  2 
  ; 
  they 
  

   maybe 
  quite 
  independent, 
  or 
  may 
  be 
  chosen 
  to 
  give 
  achro- 
  

   matism 
  between 
  any 
  two 
  colours, 
  or 
  to 
  give 
  a 
  desired 
  

   chromatic 
  error, 
  or 
  to 
  satisfy 
  some 
  other 
  condition 
  not 
  

   stated 
  ; 
  they 
  may 
  be 
  of 
  like 
  or 
  unlike 
  signs. 
  If 
  a 
  ray 
  incident 
  

   on 
  the 
  system 
  is 
  converging 
  toward 
  a 
  point 
  beyond 
  the 
  

   system, 
  the 
  reciprocal 
  of 
  its 
  intersection-distance 
  will 
  give 
  

   the 
  initial 
  convergence, 
  u 
  x 
  . 
  

  

  After 
  the 
  ray 
  has 
  passed 
  through 
  the 
  first 
  surface 
  the 
  quantity 
  

   Uy 
  becomes 
  n/; 
  and 
  as 
  the 
  thickness 
  of 
  the 
  lens 
  is 
  neglected, 
  

   Ui=u 
  2 
  , 
  and 
  so 
  on. 
  All 
  these 
  c's, 
  it's 
  and 
  p's 
  are 
  to 
  be 
  thought 
  

   of 
  as 
  " 
  angles 
  per 
  unit 
  height 
  of 
  incidence." 
  The 
  excess 
  of 
  

  

  