﻿Focal 
  Plane 
  of 
  a 
  Telescope 
  with 
  Circular 
  Aperture. 
  7 
  

  

  3. 
  Different 
  Expressions 
  for 
  J 
  2 
  (x) 
  + 
  J^x). 
  

  

  Different 
  expressions 
  can 
  be 
  found 
  for 
  y 
  ; 
  for 
  small 
  values 
  

   of 
  x, 
  we 
  may 
  expand 
  it 
  in 
  powers 
  of 
  a, 
  by 
  means 
  of 
  well- 
  

   known 
  definite 
  integrals 
  : 
  — 
  - 
  

  

  1 
  (V 
  

  

  J 
  2 
  (#) 
  = 
  — 
  1 
  J 
  (2# 
  sin 
  to) 
  dco, 
  

  

  ""Jo 
  

  

  1 
  (V 
  

  

  J 
  1 
  2 
  (,'r) 
  = 
  — 
  1 
  J 
  (2^ 
  sin 
  w) 
  cos 
  2a> 
  <ia>, 
  

  

  ^ 
  Jo 
  

  

  2 
  f 
  v 
  

   Jo 
  2 
  (#) 
  + 
  Ji 
  2 
  (#) 
  = 
  — 
  1 
  J 
  (2cr 
  sin 
  a)) 
  cos 
  2 
  gj 
  da). 
  

  

  By 
  expanding 
  J 
  (2# 
  cos 
  o>) 
  in 
  powers 
  of 
  2 
  a? 
  cos 
  <w, 
  and 
  

   integrating, 
  we 
  obtain 
  

  

  J 
  2(V) 
  + 
  J, 
  2 
  (V)= 
  2 
  (-^y 
  2 
  ^) 
  . 
  .^» 
  . 
  . 
  (A) 
  

  

  ow 
  lW 
  2 
  2w 
  + 
  i 
  (n(n))- 
  :5 
  n^ 
  + 
  i) 
  

  

  The 
  ascending 
  series 
  converges 
  very 
  slowly 
  for 
  large 
  values 
  

   of 
  x 
  ; 
  we 
  may 
  conveniently 
  employ 
  the 
  following 
  semi- 
  

   convergent 
  series, 
  which 
  can 
  be 
  easily 
  found 
  from 
  the 
  corre- 
  

   sponding 
  expansions 
  for 
  J 
  (x) 
  and 
  Ji(#). 
  Thus, 
  

  

  t 
  9/ 
  x 
  1 
  /i 
  1 
  • 
  -» 
  cos 
  2^7 
  5 
  sin 
  2#\ 
  

  

  T 
  0/ 
  x 
  1 
  /., 
  3 
  1 
  . 
  a 
  3 
  cos 
  2<r 
  3 
  sin 
  2x\ 
  

  

  J/W 
  = 
  — 
  I 
  1 
  + 
  c 
  - 
  2 
  — 
  sm2o?— 
  -: 
  ^j. 
  o 
  - 
  ), 
  

  

  1 
  v 
  ' 
  irx\ 
  b 
  x 
  2 
  4 
  x 
  62 
  x 
  2 
  J 
  

  

  whence 
  

  

  T2MlT2M 
  2 
  / 
  1 
  cos 
  2,2? 
  sin 
  2#\ 
  ,,,. 
  

  

  The 
  above 
  series 
  (B) 
  is 
  rapidly 
  convergent, 
  and 
  can 
  be 
  

   conveniently 
  used 
  for 
  values 
  of 
  x 
  greater 
  than 
  the 
  first 
  root 
  

   #1 
  = 
  3*8317 
  of 
  Ji(a?) 
  =0 
  ; 
  at 
  the 
  last-mentioned 
  value 
  of 
  x, 
  the 
  

   number 
  obtained 
  for 
  J 
  2 
  (x) 
  + 
  Ji 
  2 
  (#) 
  will 
  be 
  accurate 
  to 
  the 
  

   fourth 
  decimal 
  place. 
  

  

  In 
  the 
  neighbourhood 
  of 
  the 
  inflexion-points 
  of 
  the 
  first 
  

   set, 
  y 
  remains 
  nearly 
  constant 
  ; 
  we 
  can 
  thus 
  expand 
  

   Jo 
  2 
  (#) 
  + 
  Ji 
  2 
  (#) 
  in 
  Maclaurin's 
  series. 
  Denoting 
  the 
  roots 
  of 
  

   Ji(a?)=0 
  by 
  a? 
  , 
  x 
  v 
  , 
  and 
  putting 
  

  

  y„ 
  = 
  J 
  {x 
  n 
  ), 
  £ 
  = 
  x~x 
  n 
  , 
  

  

  we 
  shall 
  obtain 
  the 
  following 
  series 
  for 
  y 
  in 
  the 
  neighbour- 
  

   hood 
  of 
  the 
  point 
  x 
  n 
  , 
  y 
  n 
  : 
  — 
  

  

  jr- 
  VW 
  [i- 
  ^ 
  • 
  f 
  + 
  g-£ 
  -4(3 
  -4)Ji- 
  + 
  . 
  .'.]. 
  (C) 
  

  

  