﻿a 
  Spherical 
  Gaseous 
  Nebula. 
  13 
  

  

  the 
  two 
  cases 
  /c=l, 
  and 
  /e=5 
  (k 
  = 
  2 
  and 
  /c 
  = 
  l*2). 
  For 
  /e 
  = 
  l, 
  

   the 
  solution, 
  in 
  the 
  present 
  notation, 
  is 
  — 
  

  

  ©!(#)=;* 
  sin- 
  (7); 
  

  

  a 
  result 
  which 
  was 
  first 
  given 
  by 
  Hitter. 
  For 
  /e=5, 
  the 
  

   solution 
  is 
  

  

  @ 
  ^)=75ra) 
  (8) 
  - 
  

  

  § 
  7. 
  The 
  method 
  of 
  obtaining 
  numerical 
  solutions 
  of 
  

   equation 
  (1) 
  which 
  has 
  been 
  adopted 
  throughout 
  the 
  present 
  

   paper, 
  is 
  derived 
  from 
  that 
  indicated 
  by 
  Lord 
  Kelvin 
  on 
  

   page 
  291 
  o£ 
  his 
  paper 
  to 
  the 
  Philosophical 
  Magazine, 
  March 
  

   1887, 
  referred 
  to 
  in 
  § 
  2 
  above. 
  An 
  arbitrary 
  trial 
  curve, 
  t 
  , 
  

  

  fulfilling 
  the 
  initial 
  conditions 
  t 
  = 
  A; 
  — 
  =A'; 
  at 
  w=za 
  ; 
  is 
  

  

  taken 
  for 
  t. 
  From 
  this 
  curve, 
  t 
  Q 
  , 
  a 
  curve 
  representing 
  

  

  — 
  -4, 
  or 
  -j\ 
  is 
  obtained 
  by 
  direct 
  calculation. 
  One 
  inte- 
  

   so 
  ax 
  

  

  gration 
  performed 
  on 
  this 
  calculated 
  curve 
  gives 
  the 
  means 
  

  

  of 
  drawing 
  — 
  ^ 
  ; 
  and 
  the 
  curve 
  representing 
  £ 
  l5 
  which 
  is 
  

  

  ' 
  dt 
  

  

  obtained 
  by 
  integrating 
  — 
  , 
  is 
  then 
  found 
  to 
  be 
  a 
  closer 
  

  

  approximation 
  to 
  the 
  true 
  curve 
  for 
  t 
  than 
  the 
  curve 
  t 
  chosen 
  

   arbitrarily. 
  This 
  process 
  may 
  be 
  repeated 
  to 
  obtain 
  t 
  2 
  , 
  using 
  

   the 
  curve 
  t 
  x 
  as 
  a 
  new 
  trial 
  curve 
  for 
  t 
  : 
  and 
  so 
  on. 
  When 
  

   U 
  differs 
  very 
  little 
  from 
  ti-i, 
  U 
  may 
  be 
  regarded 
  as 
  a 
  very 
  

   close 
  approximation 
  to 
  the 
  true 
  curve 
  for 
  t. 
  

  

  § 
  8. 
  It 
  was 
  found 
  that 
  the 
  best 
  way 
  to 
  carry 
  out 
  this 
  plan 
  

   of 
  obtaining 
  a 
  numerical 
  solution 
  by 
  means 
  of 
  two 
  successive 
  

   integrations 
  was 
  to 
  choose 
  the 
  interval 
  of 
  integration, 
  from 
  

  

  any 
  point 
  at 
  which 
  t 
  and 
  — 
  were 
  known, 
  sufficiently 
  small 
  

  

  a 
  on 
  

  

  that 
  the 
  trial 
  value 
  chosen 
  for 
  t, 
  for 
  the 
  point 
  at 
  the 
  end 
  of 
  

   the 
  interval, 
  could 
  be 
  determined 
  with 
  any 
  degree 
  of 
  accuracy 
  

   required, 
  by 
  means 
  of 
  numerical 
  differences 
  of 
  the 
  values 
  of 
  t 
  

   already 
  determined 
  for 
  the 
  end 
  points 
  of 
  preceding 
  intervals. 
  

   The 
  arbitrary 
  trial 
  curve 
  is 
  in 
  this 
  case 
  a 
  curve 
  which 
  

   coincides 
  with 
  the 
  true 
  curve 
  in 
  each 
  interval 
  preceding 
  the 
  

   one 
  considered, 
  and 
  in 
  this 
  interval 
  it 
  closely 
  approximates 
  to 
  

   the 
  true 
  curve. 
  It 
  has 
  been 
  found 
  that 
  one 
  application 
  of 
  the 
  

   process 
  of 
  double 
  integration 
  to 
  this 
  trial 
  curve 
  gives 
  t 
  and 
  

  

  