﻿round 
  the 
  Focus 
  of 
  a 
  Lens, 
  at 
  Various 
  Apertures. 
  39 
  

  

  When 
  the 
  first 
  cusp 
  is 
  exceeded, 
  the 
  arc 
  length 
  I 
  loses 
  

   its 
  simple 
  physical 
  meaning 
  *. 
  But 
  this 
  will 
  occur 
  the 
  

   farther 
  away 
  from 
  the 
  focus, 
  the 
  smaller 
  the 
  angular 
  semi- 
  

   aperture 
  \/u. 
  

  

  If 
  the 
  wave-surface 
  which 
  we 
  are 
  projecting, 
  so 
  to 
  speak, 
  

   on 
  the 
  sphere 
  §-, 
  is 
  produced 
  by 
  a 
  centred 
  lens 
  or 
  lens 
  system 
  

   from, 
  say, 
  a 
  plane, 
  axially 
  incident, 
  perfect 
  wave, 
  and 
  if 
  the 
  

   centre 
  is 
  the 
  focus 
  of 
  that 
  lens, 
  then, 
  as 
  has 
  already 
  been 
  

   mentioned, 
  tj 
  will 
  be 
  of 
  the 
  form 
  bd 
  i 
  + 
  b'6 
  5 
  + 
  . 
  . 
  ., 
  or 
  in 
  

   terms 
  of 
  u, 
  

  

  v 
  = 
  bu 
  2 
  + 
  b'u* 
  + 
  (14) 
  

  

  The 
  constant 
  coefficients 
  6, 
  b', 
  etc. 
  will 
  depend 
  on 
  the 
  

   individual 
  properties 
  of 
  the 
  lens. 
  

  

  The 
  above 
  generalities 
  will 
  be 
  made 
  plain 
  presently 
  by 
  

   working 
  out 
  in 
  detail 
  the 
  case 
  of 
  the 
  most 
  simple 
  lens. 
  

   Other 
  cases 
  can 
  be 
  treated 
  on 
  similar 
  lines. 
  

  

  The 
  Piano- 
  Convex 
  Lens. 
  

   Let 
  a 
  beam 
  of 
  parallel, 
  cophasal, 
  rays 
  of 
  monochromatic 
  

   light 
  of 
  wave-length 
  \ 
  impinge 
  upon 
  the 
  flat 
  face 
  of 
  a 
  plano- 
  

   convex 
  lens 
  of 
  refractive 
  index 
  n. 
  Let 
  r 
  be 
  the 
  radius 
  of 
  

   curvature 
  of 
  the 
  convex 
  (spherical) 
  face. 
  Let 
  our 
  centre 
  

   be 
  the 
  focus, 
  and 
  i£ 
  = 
  ON 
  its 
  shortest 
  distance 
  from 
  the 
  

  

  Fig. 
  2. 
  

  

  

  

  N* 
  L 
  

  

  

  

  

  > 
  

  

  ^\"S<r 
  

  

  

  

  b 
  

  

  ! 
  jN 
  

  

  

  £=-JL 
  

  

  

  

  /r 
  \ 
  

  

  

  0* 
  

  

  convex 
  face 
  of 
  the 
  lens. 
  As 
  our 
  reference 
  surface 
  5 
  (hitherto 
  

   so 
  called) 
  let 
  us 
  take 
  the 
  sphere 
  of 
  radius 
  R 
  and 
  centre 
  

   (fig. 
  2). 
  A 
  ray 
  incident 
  at 
  height 
  h 
  will 
  be 
  refracted 
  at 
  A, 
  

  

  * 
  If 
  x 
  x 
  < 
  ot\/u 
  < 
  x 
  2 
  , 
  and 
  if 
  l 
  x 
  be 
  the 
  arc 
  from 
  the 
  origin 
  to 
  the 
  first 
  

   cusp, 
  then 
  I, 
  in 
  (11) 
  and 
  (13), 
  is 
  to 
  be 
  replaced 
  by 
  l 
  x 
  minus 
  the 
  arc 
  length 
  

   from 
  the 
  first 
  cusp 
  to 
  the 
  point 
  u 
  ; 
  and 
  so 
  on. 
  

  

  