﻿Gravitational 
  Matter 
  in 
  any 
  part 
  of 
  the 
  Universe. 
  7 
  

  

  homogeneous 
  gravitational 
  matter 
  given 
  at 
  rest 
  within 
  a 
  

   spherical 
  surface 
  and 
  left 
  to 
  fall 
  inwards, 
  the 
  augmenting 
  

   density 
  remains 
  homogeneous, 
  and 
  the 
  time 
  of 
  shrinkage 
  to 
  

   any 
  stated 
  proportion 
  of 
  the 
  initial 
  radius 
  is 
  inversely 
  as 
  the 
  

   square 
  root 
  of 
  the 
  density. 
  

  

  To 
  apply 
  this 
  result 
  to 
  the 
  supposed 
  spherical 
  universe 
  of 
  

   radius 
  3'09.10 
  16 
  kilometres, 
  and 
  mass 
  equal 
  to 
  a 
  thousand 
  

   million 
  times 
  the 
  mass 
  of 
  our 
  sun, 
  we 
  find 
  the 
  gravitational 
  

   attraction 
  on 
  a 
  body 
  at 
  its 
  surface 
  gives 
  acceleration 
  of 
  

   1*37. 
  10 
  -13 
  kilometres 
  per 
  second 
  per 
  second. 
  This 
  therefore 
  

  

  is 
  the 
  value 
  of 
  -~ 
  # 
  , 
  with 
  one 
  second 
  as 
  the 
  unit 
  of 
  time 
  

  

  and 
  one 
  kilometre 
  as 
  the 
  unit 
  of 
  distance 
  ; 
  and 
  we 
  find 
  

   T=52*8.10 
  13 
  seconds 
  = 
  16*8 
  million 
  years. 
  Thus 
  our 
  formulas 
  

   become 
  

  

  ir 
  2 
  = 
  l-37.10- 
  13 
  ^ 
  

  

  (?-*) 
  

  

  giving 
  

   and 
  

  

  »=5-23.10-y%,g<Ul) 
  

  

  whence, 
  when 
  sin 
  is 
  very 
  small, 
  

  

  / 
  40 
  3 
  \ 
  

  

  Let 
  now, 
  for 
  example, 
  «£ 
  = 
  3*09.10 
  16 
  kilometres, 
  and 
  — 
  =10 
  7 
  ; 
  

  

  x 
  

  

  and, 
  therefore, 
  sin 
  0=0 
  = 
  3*16. 
  10~ 
  4 
  ; 
  whence, 
  v=291,000 
  

  

  kilometres 
  per 
  second, 
  and 
  £ 
  = 
  T 
  — 
  7080 
  seconds 
  = 
  T 
  — 
  2 
  hours 
  

  

  approximately. 
  

  

  By 
  these 
  results 
  it 
  is 
  most 
  interesting 
  to 
  know 
  that 
  our 
  

  

  supposed 
  sphere 
  of 
  perfectly 
  compressible 
  fluid, 
  beginning 
  at 
  

  

  rest 
  with 
  density 
  1*61.10 
  -23 
  of 
  that 
  of 
  water, 
  and 
  of 
  any 
  

  

  magnitude 
  large 
  or 
  small, 
  and 
  left 
  unclogged 
  by 
  ether 
  to 
  

  

  shrink 
  under 
  the 
  influence 
  of 
  mutual 
  gravitation 
  of 
  its 
  parts, 
  

  

  would 
  take 
  nearly 
  seventeen 
  million 
  years 
  to 
  reach 
  *01G1 
  of 
  

  

  the 
  density 
  of 
  water, 
  and 
  about 
  two 
  hours 
  longer 
  to 
  shrink 
  to 
  

  

  infinite 
  density 
  at 
  its 
  centre. 
  It 
  is 
  interesting 
  also 
  to 
  know 
  

  

  that 
  if 
  the 
  initial 
  radius 
  is 
  3*09. 
  10 
  16 
  kilometres, 
  the 
  inward 
  

  

  velocity 
  of 
  the 
  surface 
  is 
  291,000 
  kilometres 
  per 
  second 
  at 
  

  

  the 
  instant 
  when 
  its 
  radius 
  is 
  3*09. 
  10 
  9 
  and 
  its 
  density 
  *0161 
  

  

  