﻿round 
  the 
  Focus 
  of 
  a 
  Lens, 
  at 
  Various 
  Apertures. 
  3? 
  

  

  arc 
  elements 
  dl 
  corresponding 
  to 
  equal 
  steps 
  du, 
  for 
  axial 
  

   points 
  P(a 
  = 
  0), 
  are 
  all 
  equal, 
  as 
  in 
  the 
  case 
  of 
  the 
  Cornu 
  

   spiral. 
  Outside 
  the 
  optical 
  axis, 
  however, 
  the 
  steps 
  dl 
  

   become 
  smaller 
  and 
  smaller 
  as 
  we 
  approach 
  the 
  first 
  zero 
  

   of 
  J 
  , 
  which 
  happens 
  the 
  sooner 
  the 
  larger 
  a. 
  At 
  the 
  

  

  same 
  time 
  the 
  radius 
  of 
  curvature 
  -7 
  dwindles 
  to 
  nothing 
  

  

  dn 
  

   (unless 
  -—■ 
  = 
  /3) 
  ; 
  at 
  the 
  apertures 
  corresponding 
  to 
  the 
  zeros 
  

  

  of 
  J 
  (u\/u) 
  the 
  P-curves 
  have 
  cusps, 
  from 
  which 
  they 
  emerge 
  

   with 
  increasing 
  steps 
  to 
  be 
  lessened 
  again 
  when 
  the 
  next 
  

   zero 
  is 
  approached, 
  and 
  so 
  on. 
  For 
  u 
  — 
  0, 
  rj 
  — 
  and, 
  therefore, 
  

   € 
  = 
  ; 
  thus, 
  all 
  P-curves 
  start 
  from 
  the 
  origin 
  tangentially 
  

   to 
  the 
  #-axis. 
  Again, 
  since, 
  in 
  all 
  cases 
  of 
  actual 
  interest, 
  

  

  t~ 
  = 
  for 
  w 
  = 
  0, 
  the 
  initial 
  curvature 
  of 
  all 
  curves 
  belonging 
  

  

  to 
  the 
  focal 
  plane 
  is 
  nil 
  ; 
  and 
  the 
  initial 
  curvature 
  for 
  any 
  other 
  

  

  point 
  P 
  is 
  k= 
  — 
  j3= 
  — 
  . 
  But 
  for 
  any 
  not 
  highly 
  corrected 
  

  

  dn 
  

   lens 
  the 
  term 
  -~ 
  in 
  (10) 
  soon 
  becomes 
  the 
  more 
  important 
  

  

  one, 
  unless 
  a 
  mounts 
  to 
  many 
  wave-lengths. 
  Thus, 
  with 
  

   the 
  exception 
  of 
  the 
  first 
  steps, 
  the 
  sloping 
  angle 
  is 
  given 
  

  

  dw 
  

   primarily 
  by 
  rj, 
  and 
  the 
  curvature 
  by 
  -=- 
  : 
  | 
  J 
  |, 
  the 
  modifi- 
  

   cations 
  due 
  to 
  an 
  axial 
  displacement 
  being 
  comparatively 
  

   small. 
  

  

  In 
  most 
  cases, 
  therefore, 
  it 
  will 
  be 
  found 
  that 
  it 
  is 
  sufficient 
  

   to 
  draw 
  in 
  detail 
  the 
  P-curves 
  for 
  the 
  focal 
  plane 
  only, 
  when 
  

   the 
  construction 
  data 
  become 
  

  

  e 
  = 
  v 
  , 
  dl=\J 
  \du, 
  k=^f 
  u 
  . 
  

  

  If 
  drjjdu 
  preserves 
  its 
  sign, 
  the 
  sense 
  of 
  the 
  windings 
  of 
  a 
  

   P-curve 
  remains 
  throughout 
  the 
  same 
  (say, 
  anticlockwise), 
  

   even 
  in 
  passing 
  through 
  a 
  cusp. 
  If 
  rj 
  passes 
  through 
  a 
  

   maximum 
  or 
  minimum 
  (and 
  J 
  gfc0), 
  the 
  curve 
  becomes 
  

   flat 
  and 
  inflected. 
  If 
  a 
  P-curve, 
  no 
  matter 
  after 
  how 
  

   many 
  windings, 
  happens 
  to 
  pass 
  again 
  through 
  the 
  origin, 
  

   the 
  light 
  at 
  P 
  * 
  is 
  extinguished, 
  and 
  while 
  the 
  curve 
  passes 
  

   on 
  (increasing 
  aperture), 
  the 
  light 
  will 
  reappear 
  there, 
  and 
  

   so 
  on. 
  A 
  good 
  check 
  at 
  any 
  stage 
  of 
  the 
  curve 
  construction 
  

   may 
  be 
  to 
  measure 
  the 
  whole 
  length 
  I 
  of 
  the 
  path 
  already 
  

   covered 
  and 
  to 
  compare 
  it 
  with 
  its 
  correct 
  length, 
  which 
  can 
  

  

  * 
  i. 
  e., 
  along 
  the 
  circle 
  through 
  P, 
  centred 
  upon 
  and 
  normal 
  to 
  the 
  

   optical 
  axis. 
  

  

  