﻿a 
  Spherical 
  Gaseous 
  Nebula. 
  15 
  

  

  limit 
  away 
  from 
  it. 
  From 
  similar 
  considerations 
  it 
  may 
  be 
  

   judged 
  that 
  when 
  t 
  has 
  become 
  less 
  than 
  its 
  true 
  value, 
  in 
  the 
  

   succeeding 
  intervals 
  it 
  tends 
  to 
  return 
  towards 
  its 
  true 
  value. 
  

   § 
  10. 
  The 
  numerical 
  values 
  of 
  the 
  Homer 
  Lane 
  Function 
  

   S(x) 
  with 
  its 
  differential 
  coefficient 
  ®'(V)> 
  in 
  the 
  interval 
  

   from 
  x 
  = 
  00 
  to 
  x=q, 
  given 
  in 
  Tables 
  I. 
  . 
  . 
  . 
  IV., 
  and 
  the 
  

   values 
  of 
  the 
  Boylean 
  Function 
  "fy(x) 
  y 
  and 
  the 
  function 
  

   ^'(x)/^(x), 
  in 
  the 
  interval 
  from 
  x 
  = 
  '25, 
  to 
  a' 
  = 
  'l, 
  given 
  in 
  

   Table 
  V., 
  have 
  all 
  been 
  obtained 
  by 
  the 
  method 
  of 
  § 
  8. 
  In 
  

   each 
  of 
  the 
  Homer 
  Lane 
  Functions, 
  a 
  beginning 
  of 
  the 
  calcu- 
  

   lation 
  was 
  made 
  from 
  the 
  following 
  approximate 
  equation 
  : 
  — 
  

  

  *<•>¥- 
  «?+ 
  B5?- 
  • 
  • 
  • 
  < 
  12 
  )> 
  

  

  easily 
  derived 
  from 
  equations 
  (2) 
  and 
  (4) 
  above. 
  From 
  

   ^ 
  = 
  00 
  to 
  x 
  = 
  '25, 
  the 
  Boylean 
  Function 
  was 
  calculated 
  by 
  a 
  

   method 
  described 
  below. 
  

  

  After 
  Tables 
  I. 
  . 
  . 
  . 
  IY. 
  had 
  been 
  completed, 
  as 
  it 
  was 
  still 
  

   desirable 
  to 
  be 
  able 
  to 
  verify 
  the 
  results 
  obtained 
  by 
  the 
  

   step-by-step 
  process 
  at 
  some 
  point 
  close 
  to 
  the 
  final 
  point, 
  q, 
  

   and 
  as 
  the 
  labour 
  of 
  calculating 
  successive 
  terms 
  of 
  the 
  

   series 
  (2) 
  soon 
  becomes 
  very 
  great, 
  while 
  the 
  number 
  of 
  the 
  

   terms 
  required 
  to 
  give 
  a 
  sufficiently 
  good 
  result 
  also 
  becomes 
  

   greater 
  and 
  greater 
  as 
  x 
  diminishes, 
  the 
  form 
  of 
  expansion 
  

   given 
  in 
  § 
  11 
  below 
  was 
  tried, 
  and 
  found 
  to 
  be 
  useful. 
  

  

  § 
  11. 
  Assume 
  as 
  a 
  solution 
  of 
  equation 
  (1), 
  

  

  S 
  K 
  {x) 
  = 
  

  

  ( 
  1+ 
  ^ 
  4 
  -^ 
  +etc 
  .y 
  • 
  • 
  (i3). 
  

  

  V 
  & 
  xA 
  x 
  Q 
  J 
  

  

  We 
  can 
  write 
  %" 
  K 
  (x) 
  in 
  either 
  of 
  the 
  following 
  forms 
  : 
  — 
  

  

  (n-3+5+3+-*r 
  ' 
  

  

  2.3.ai» 
  

  

  >3+3H 
  (y-3+S+3+*-r 
  

  

  If 
  we 
  now 
  choose 
  n 
  in 
  the 
  first 
  form, 
  so 
  that 
  (n 
  + 
  2)=/en, 
  and 
  

  

  tor 
  to 
  , 
  we 
  obtain 
  

  

  x* 
  

  

  © 
  // 
  K 
  ( 
  A 
  .)=_[@ 
  (C 
  (^)]7^. 
  

  

  equate 
  the 
  numerator 
  to 
  4 
  , 
  we 
  obtain 
  

  

  