﻿a 
  Spherical 
  Gaseous 
  Nebula. 
  17 
  

  

  We 
  may, 
  in 
  this 
  case 
  also, 
  write 
  * 
  2 
  J 
  in 
  the 
  two 
  

  

  forms 
  :— 
  dx 
  

  

  /2.3.a! 
  , 
  4.5.a 
  2 
  , 
  , 
  \L 
  , 
  a 
  x 
  , 
  a 
  2 
  , 
  , 
  \ 
  , 
  /2a 
  x 
  4a 
  2 
  . 
  . 
  

  

  - 
  B 
  h^ 
  + 
  ~ 
  +ete 
  ') 
  ( 
  1+ 
  5 
  + 
  * 
  +etc 
  -) 
  + 
  "b> 
  +^ 
  +etc 
  - 
  

  

  2.3.aiffl 
  

  

  3* 
  

  

  *( 
  

  

  (^S+S+SW 
  

  

  a?6 
  a? 
  8 
  

  

  )(i+^+?+ 
  

  

  Xi 
  

  

  \ 
  (2a 
  Y 
  4a 
  2 
  

   / 
  \x* 
  x 
  a 
  

  

  *)' 
  

  

  \ 
  X 
  Z 
  ' 
  X 
  

  

  By 
  taking 
  72 
  = 
  2 
  in 
  the 
  first 
  of 
  these 
  forms, 
  and 
  n=l 
  in 
  the 
  

  

  second, 
  and 
  proceeding 
  exactly 
  as 
  in 
  § 
  11, 
  we 
  obtain 
  two 
  

  

  expressions 
  for 
  *P(x) 
  in 
  the 
  form 
  (17), 
  the 
  coefficients 
  « 
  1? 
  a 
  2 
  , 
  

  

  a 
  3 
  , 
  etc. 
  being 
  again 
  determined 
  from 
  the 
  series 
  of 
  equations 
  

  

  (14), 
  when 
  n 
  = 
  2, 
  and 
  from 
  equations 
  (15), 
  when 
  n 
  = 
  l. 
  

  

  § 
  13. 
  At 
  present 
  only 
  the 
  numerical 
  values 
  given 
  by 
  the 
  

  

  above 
  solutions 
  are 
  of 
  importance 
  ; 
  and 
  for 
  this 
  purpose, 
  the 
  

  

  2 
  

   expression 
  (13) 
  with 
  n 
  = 
  has 
  been 
  used 
  to 
  calculate 
  

  

  the 
  Homer 
  Lane 
  Functions, 
  and 
  the 
  expression 
  (17) 
  with 
  n 
  = 
  2 
  

   to 
  calculate 
  the 
  Boylean 
  Function. 
  The 
  following 
  table 
  gives 
  

   the 
  logarithms 
  of 
  several 
  of 
  the 
  coefficients 
  a 
  1? 
  « 
  2 
  , 
  etc. 
  for 
  the 
  

   functions 
  given 
  in 
  Tables 
  I. 
  ... 
  V. 
  ; 
  the 
  letter 
  n 
  indicating 
  

   that 
  the 
  term 
  is 
  negative 
  : 
  — 
  

  

  K 
  

  

  1-5 
  

  

  2-5 
  

  

  3 
  

  

  4 
  

  

  CO 
  

  

  log'ui 
  

  

  2-61979 
  

  

  1-09691 
  

  

  1-22185 
  

  

  1-39794 
  

  

  2-92082 
  

  

  log 
  a 
  2 
  

  

  3-08468 
  

  

  3-41567 
  

  

  3-44370 
  

  

  3-31876 
  

  

  4-84164% 
  

  

  loga 
  3 
  

  

  5-81361 
  

  

  5-94369 
  

  

  75-82045 
  

  

  "6-91736 
  

  

  5-13921 
  

  

  log 
  GT4 
  

  

  6-59587 
  

  

  6-45524 
  

  

  6-10924 
  

  

  7-23612 
  

  

  7-61229 
  n 
  

  

  log 
  a 
  5 
  . 
  

  

  739883 
  

  

  8-96928 
  

  

  8-46991 
  

  

  9-5540 
  » 
  

  

  8-19003 
  

  

  log 
  a 
  G 
  

  

  8-21416 
  

  

  9-49350 
  

  

  10-72486 
  

  

  U-8517 
  

  

  10-83708 
  n 
  

  

  log 
  a 
  7 
  

  

  9-03834 
  

  

  10-01706 
  

  

  H-19035 
  

  

  n 
  

  

  11-5332 
  

  

  log 
  as 
  

  

  11-86917 
  

  

  12-54628 
  

  

  14-84984 
  

  

  

  n 
  

  

  log 
  a 
  9 
  

  

  12-70515 
  

  

  13-07363 
  

  

  14-26996 
  

  

  

  

  log 
  a 
  w 
  

  

  13-54520 
  

  

  15-60715 
  

  

  16-9440 
  n 
  

  

  

  

  log 
  a 
  n 
  

  

  14-38854 
  

  

  16- 
  13498 
  

  

  i'7-9204 
  

  

  

  

  log 
  a 
  12 
  

  

  15-23460 
  

  

  18-67438 
  

  

  n 
  

  

  

  

  log 
  a 
  13 
  

  

  

  19-1967 
  

  

  

  

  

  log 
  a 
  u 
  

  

  

  21-7511 
  

  

  

  

  

  log 
  a 
  X5 
  

  

  

  22-2473 
  

  

  

  

  

  log 
  «16 
  

  

  

  24-8548 
  

  

  

  

  

  log 
  a 
  17 
  

  

  

  25-4061 
  

  

  

  

  

  log 
  a 
  18 
  

  

  

  27-6267 
  

  

  

  1 
  

  

  

  Phil. 
  Mag. 
  S. 
  6. 
  Vol. 
  16. 
  No. 
  91. 
  July 
  1908. 
  

  

  

  

  