﻿12 
  Lord 
  Kelvin 
  : 
  The 
  Problem 
  of 
  

  

  These 
  terms 
  are 
  sufficient 
  to 
  give 
  a 
  satisfactory 
  determination 
  

   of 
  t 
  for 
  all 
  values 
  of 
  a 
  equal 
  to 
  or 
  greater 
  than 
  unity 
  ; 
  that 
  

   is, 
  at 
  all 
  points 
  in 
  the 
  gas 
  within 
  distance 
  <r 
  from 
  the 
  centre 
  

   of 
  the 
  sphere, 
  where 
  the 
  radius 
  of 
  the 
  boundary 
  is 
  a/q, 
  and 
  

   q 
  is 
  the 
  value 
  of 
  x 
  for 
  which 
  @ 
  K 
  (#) 
  = 
  0. 
  

  

  The 
  labour 
  of 
  calculating 
  additional 
  terms 
  of 
  these 
  series 
  

   being 
  very 
  great, 
  and 
  no 
  great 
  precision 
  being 
  necessary, 
  

   Homer 
  Lane 
  merely 
  employed 
  a 
  step-by-step 
  process, 
  involv- 
  

   ing 
  the 
  use 
  of 
  numerical 
  differences, 
  to 
  obtain 
  approximately 
  

   the 
  value 
  of 
  t 
  at 
  points 
  whose 
  distance 
  from 
  the 
  centre 
  of 
  the 
  

   sphere 
  was 
  greater 
  than 
  that 
  corresponding 
  to 
  on 
  unity 
  in 
  

   equations 
  (5) 
  and 
  (6). 
  When 
  a 
  fairly 
  small 
  value 
  of 
  t 
  had 
  

   been 
  reached 
  by 
  this 
  method, 
  he 
  was 
  able 
  to 
  complete 
  the 
  

   calculation 
  as 
  far 
  as 
  t 
  = 
  (x=q) 
  by 
  means 
  of 
  approximate 
  

   formulas 
  which 
  can 
  be 
  derived 
  in 
  a 
  manner 
  similar 
  to 
  that 
  

   described 
  above. 
  

  

  § 
  4, 
  With 
  a 
  view 
  to 
  obtaining 
  greater 
  accuracy 
  in 
  the 
  

   results 
  for 
  monatomic 
  gases, 
  Mr. 
  T. 
  J. 
  J. 
  See 
  has 
  extended 
  

   Homer 
  Lane's 
  series 
  (5) 
  as 
  far 
  as 
  the 
  term 
  containing 
  a? 
  20 
  ; 
  

   and 
  with 
  the 
  aid 
  of 
  additional 
  terms, 
  obtained 
  by 
  means 
  of 
  

   logarithmic 
  differences 
  of 
  preceding 
  terms, 
  he 
  has 
  calculated 
  

   the 
  values 
  of 
  t, 
  p, 
  m, 
  etc., 
  at 
  a 
  place 
  very 
  close 
  to 
  the 
  

   boundary 
  of 
  the 
  gas. 
  From 
  this 
  he 
  has 
  been 
  able 
  to 
  find 
  

   with 
  great 
  accuracy 
  the 
  radius 
  of 
  the 
  spherical 
  boundary, 
  

   and 
  the 
  total 
  mass 
  of 
  gas, 
  corresponding 
  to 
  the 
  Homer 
  Lane 
  

   Function 
  ©1-5(0;) 
  (see 
  § 
  17). 
  These 
  results 
  are 
  published 
  in 
  

   a 
  paper 
  entitled 
  " 
  Researches 
  on 
  the 
  physical 
  constitution 
  of 
  

   the 
  heavenly 
  bodies 
  " 
  (Astr. 
  Nachr. 
  No. 
  1053, 
  Bd. 
  169, 
  Nov. 
  

   1905). 
  They 
  were 
  found 
  after 
  Table 
  I., 
  on 
  page 
  20 
  of 
  the 
  

   present 
  paper, 
  had 
  been 
  completed 
  by 
  the 
  entirely 
  different 
  

   method 
  given 
  below, 
  and 
  they 
  are 
  a 
  confirmation 
  of 
  its 
  

   usefulness. 
  

  

  § 
  5. 
  Eight 
  years 
  after 
  the 
  publication 
  of 
  Homer 
  Lane's 
  

   famous 
  paper, 
  the 
  problem 
  of 
  the 
  convective 
  equilibrium 
  of 
  

   a 
  spherical 
  mass 
  of 
  gas 
  under 
  its 
  own 
  gravitation 
  only 
  was 
  

   dealt 
  with 
  very 
  fully 
  by 
  A. 
  Ritter, 
  in 
  a 
  series 
  of 
  papers 
  

   entitled 
  " 
  Untersuchungen 
  iiber 
  die 
  hohe 
  der 
  Atmosphare 
  

   und 
  die 
  Constitution 
  gasformiger 
  W 
  r 
  eltkorper," 
  published 
  in 
  

   Wiedemann's 
  Annalen, 
  1878-1882. 
  Numerical 
  solutions 
  of 
  

   equation 
  (1) 
  are 
  given 
  for 
  the 
  following 
  values 
  of 
  k, 
  1*5, 
  2, 
  

   2*44, 
  3, 
  4, 
  5 
  ; 
  these 
  solutions 
  being 
  obtained 
  wholly 
  by 
  a 
  

   graphical 
  process, 
  similar 
  to 
  the 
  process 
  described 
  in 
  § 
  7 
  

   below. 
  

  

  § 
  6. 
  Professor 
  Schuster, 
  in 
  a 
  short 
  paper 
  to 
  the 
  British 
  

   Association 
  at 
  Southport 
  in 
  1883, 
  pointed 
  out 
  that 
  it 
  was 
  

   possible 
  to 
  obtain 
  solutions 
  of 
  equation 
  (1) 
  in 
  finite 
  terms 
  in 
  

  

  