﻿6 
  Lord 
  Kelvin 
  on 
  the 
  Clustering 
  of 
  

  

  total 
  eclipse 
  of 
  the 
  sun 
  by 
  a 
  natural 
  cloud 
  of 
  water 
  spherules, 
  

   or 
  by 
  the 
  cloud 
  of 
  smoke 
  from 
  the 
  funnel 
  of 
  a 
  steamer. 
  

  

  Let 
  now 
  all 
  the 
  matter 
  in 
  our 
  supposed 
  universe 
  be 
  reduced 
  

   to 
  atoms 
  (literally 
  brought 
  back 
  to 
  its 
  probable 
  earliest 
  

   condition). 
  Through 
  a 
  sphere 
  of 
  radius 
  r 
  let 
  atoms 
  be 
  

   distributed 
  uniformly 
  in 
  respect 
  to 
  gravitational 
  quality. 
  

   It 
  is 
  to 
  be 
  understood 
  that 
  the 
  condition 
  ' 
  uniformly 
  ' 
  is 
  

   fulfilled 
  if 
  equivoluminal 
  globular 
  or 
  cubic 
  portions, 
  small 
  

   in 
  comparison 
  with 
  the 
  whole 
  sphere, 
  but 
  large 
  enough 
  to 
  

   contain 
  large 
  numbers 
  of 
  the 
  atoms, 
  contain 
  equal 
  total 
  

   masses, 
  reckoned 
  gravitationally, 
  whether 
  the 
  atoms 
  themselves 
  

   are 
  of 
  equal 
  or 
  unequal 
  masses, 
  or 
  of 
  similar 
  or 
  dissimilar 
  

   chemical 
  qualities. 
  As 
  long 
  as 
  this 
  condition 
  is 
  fulfilled, 
  

   each 
  atom 
  experiences 
  very 
  approximately 
  the 
  same 
  force 
  as 
  

   if 
  the 
  whole 
  matter 
  were 
  infinitely 
  fine-grained, 
  that 
  is 
  to 
  say, 
  

   utterly 
  homogeneous. 
  

  

  Let 
  us 
  therefore 
  begin 
  with 
  a 
  uniform 
  sphere 
  of 
  matter 
  of 
  

   density 
  p, 
  gravitational 
  reckoning, 
  with 
  no 
  mutual 
  forces 
  

   except 
  gravitation 
  between 
  its 
  parts, 
  given 
  with 
  every 
  part 
  at 
  

   rest 
  at 
  the 
  initial 
  instant 
  ; 
  and 
  let 
  it 
  be 
  required 
  to 
  find 
  the 
  

   subsequent 
  motion. 
  Imagining 
  the 
  whole 
  divided 
  into 
  

   infinitely 
  thin 
  concentric 
  spherical 
  shells, 
  we 
  see 
  that 
  every 
  

   one 
  of 
  them 
  falls 
  inwards, 
  as 
  if 
  attracted 
  by 
  the 
  whole 
  mass 
  

   within 
  it 
  collected 
  at 
  the 
  centre. 
  Hence 
  our 
  problem 
  is 
  

   reduced 
  to 
  the 
  well 
  known 
  students' 
  exercise 
  of 
  finding 
  the 
  

   rectilinear 
  motion 
  of 
  a 
  particle 
  attracted 
  according 
  to 
  the 
  

   inverse 
  square 
  of 
  the 
  distance 
  from 
  a 
  fixed 
  point. 
  Let 
  # 
  be 
  

  

  the 
  initial 
  distance, 
  — 
  k~ 
  %o* 
  the 
  attracting 
  mass, 
  v 
  and 
  x 
  

  

  velocity 
  and 
  distance 
  from 
  the 
  centre 
  at 
  time 
  t. 
  The 
  solution 
  

   of 
  the 
  problem 
  for 
  the 
  time 
  during 
  which 
  the 
  particle 
  is 
  

   falling 
  towards 
  the 
  centre 
  is 
  

  

  and 
  

  

  <V£(F 
  9+ 
  *^W4[»-"('-^)] 
  

  

  where 
  6 
  denotes 
  the 
  acute 
  angle 
  whose 
  sine 
  is 
  a 
  / 
  — 
  . 
  This 
  

  

  V 
  OSq 
  

  

  shows 
  that 
  the 
  time 
  of 
  falling 
  through 
  any 
  proportion 
  of 
  the 
  

   initial 
  distance 
  is 
  the 
  same 
  whatever 
  be 
  the 
  initial 
  distance 
  ; 
  

   and 
  that 
  the 
  time 
  (which 
  we 
  shall 
  denote 
  by 
  T) 
  of 
  falling 
  

  

  to 
  the 
  centre 
  is 
  ^7ta 
  /~ 
  — 
  . 
  Hence 
  in 
  our 
  problem 
  of 
  

  

  