﻿a 
  Spherical 
  Gaseous 
  Nebula. 
  11 
  

  

  § 
  2. 
  Solutions 
  o£ 
  this 
  equation 
  can 
  be 
  found 
  which 
  cor- 
  

   respond 
  to 
  a 
  mass 
  of 
  gas 
  around 
  a 
  solid 
  or 
  liquid 
  nucleus. 
  

   These 
  may 
  become 
  o£ 
  interest 
  later. 
  Solutions 
  can 
  also 
  be 
  

   found 
  which 
  correspond 
  to 
  an 
  infinite 
  sphere 
  of 
  gas, 
  with 
  an 
  

   infinite 
  density 
  at 
  the 
  centre. 
  But 
  the 
  solutions 
  which 
  it 
  is 
  

   now 
  desirable 
  to 
  obtain 
  are 
  those 
  which 
  can 
  be 
  applied 
  to 
  the 
  

   case 
  of 
  a 
  spherical 
  mass 
  of 
  gas 
  which 
  has 
  a 
  finite 
  density 
  at 
  

   at 
  the 
  centre. 
  This 
  is 
  expressed 
  mathematically 
  by 
  saying 
  

  

  that 
  at 
  #=oo 
  , 
  we 
  have 
  £ 
  = 
  C, 
  -=- 
  =0. 
  

  

  ax 
  

  

  Lord 
  Kelvin 
  has 
  shown, 
  in 
  his 
  paper 
  " 
  On 
  the 
  equilibrium 
  

  

  of 
  a 
  gas 
  under 
  its 
  own 
  gravitation 
  only," 
  Phil. 
  Mag., 
  March 
  

  

  1887, 
  and 
  in 
  § 
  25 
  of 
  the 
  preceding 
  paper, 
  that, 
  when 
  any 
  

  

  complete 
  solution 
  %{x) 
  has 
  been 
  found, 
  it 
  is 
  possible 
  to 
  derive 
  

  

  from 
  it 
  a 
  general 
  solution 
  with 
  one 
  disposable 
  constant 
  C, 
  

  

  C^M 
  A'C 
  -2 
  . 
  Accordingly, 
  it 
  is 
  convenient 
  to 
  deal 
  only 
  

  

  ■ 
  ■■ 
  . 
  dt 
  

  

  with 
  the 
  particular 
  solution 
  for 
  which 
  £ 
  = 
  1; 
  -=- 
  =0 
  ; 
  at 
  

  

  x 
  = 
  co 
  , 
  denoted 
  by 
  © 
  K 
  (V), 
  and 
  called 
  the 
  Homer 
  Lane 
  

   Function. 
  

  

  § 
  3. 
  Homer 
  Lane, 
  in 
  his 
  paper 
  " 
  On 
  the 
  theoretical 
  tem- 
  

   perature 
  of 
  the 
  sun," 
  gives 
  analytical 
  solutions 
  of 
  equation 
  (1) 
  

   for 
  the 
  cases 
  & 
  = 
  1§ 
  and 
  £ 
  = 
  1*4, 
  which 
  correspond 
  to 
  a 
  mono- 
  

   tomic 
  and 
  to 
  a 
  diatomic 
  gas 
  respectively. 
  His 
  method 
  of 
  

   obtaining 
  these 
  solutions 
  is 
  equivalent 
  to 
  the 
  following. 
  

   Assume 
  

  

  i 
  = 
  0j(,.) 
  = 
  l 
  + 
  5 
  + 
  ^ 
  + 
  ^+...etc. 
  . 
  . 
  (2) 
  

  

  to 
  be 
  the 
  required 
  solution 
  of 
  (1), 
  where 
  a 
  lf 
  « 
  2 
  , 
  « 
  s 
  , 
  etc. 
  are 
  to 
  

   be 
  determined. 
  Then 
  

  

  d*® 
  K 
  2.3.«, 
  4.5.*., 
  6.7.« 
  3 
  

  

  And 
  the 
  coefficients 
  a 
  1? 
  a 
  2 
  , 
  a 
  3 
  , 
  etc. 
  can 
  be 
  determined 
  from 
  

   the 
  equation 
  — 
  

  

  \ 
  x 
  A 
  x 
  Q 
  x* 
  J 
  as* 
  

  

  The 
  solutions 
  given 
  by 
  Homer 
  Lane 
  are, 
  in 
  the 
  present 
  

   notation, 
  

  

  ®^ 
  =1 
  -i> 
  + 
  ^-uL« 
  + 
  moiJ*----- 
  etc 
  - 
  (5); 
  

  

  a 
  / 
  n 
  , 
  1 
  . 
  1 
  5 
  , 
  125 
  , 
  ... 
  

  

  *■*•)=!- 
  ^ 
  + 
  ^-—j, 
  + 
  i 
  g 
  5i 
  ^3 
  - 
  . 
  . 
  . 
  . 
  etc. 
  (6). 
  

  

  w- 
  

  

  