﻿1875.] 
  Dulong 
  and 
  Petit' 
  s 
  Law 
  of 
  Cooling, 
  467 
  

  

  any 
  arbitrary 
  function 
  of 
  x 
  from 
  x 
  = 
  to 
  x—a, 
  for 
  the 
  value 
  of 
  v, 
  

   when 
  t—0 
  : 
  

  

  v 
  temperature 
  at 
  time 
  t 
  and 
  distance 
  x 
  from 
  centre 
  of 
  

   globe, 
  

  

  a 
  the 
  radius 
  of 
  the 
  globe, 
  

  

  Jc 
  the 
  thermal 
  conductivity 
  of 
  its 
  substance, 
  

  

  c 
  the 
  thermal 
  capacity 
  per 
  unit 
  volume 
  of 
  its 
  substance, 
  

  

  E 
  the 
  thermal 
  emissivity 
  of 
  its 
  surface. 
  

  

  Taking 
  centimetre, 
  second, 
  and 
  gramme-water 
  thermal 
  units 
  for 
  the 
  

   fundamental 
  units, 
  we 
  have, 
  as 
  stated 
  above, 
  

  

  E== 
  i(ftK) 
  (rough 
  approximation) 
  ; 
  

  

  • 
  * 
  o 
  

  

  and 
  Angstrom's 
  experiments 
  gave 
  for 
  copper 
  

   &=1 
  approximately. 
  

  

  Therefore 
  

  

  Ea 
  a 
  

  

  k 
  4000 
  

  

  and 
  for 
  the 
  globe 
  of 
  4 
  centimetres 
  diameter 
  used 
  in 
  the 
  Glasgow 
  experi- 
  

   ments, 
  

  

  Eq_ 
  1 
  

   h 
  ~ 
  2000* 
  

  

  Ea 
  

  

  In 
  all 
  cases 
  in 
  which 
  — 
  is 
  small, 
  the 
  smallest 
  root 
  of 
  the 
  transcendental 
  

   fc 
  

  

  equation 
  (2) 
  is 
  approximately 
  equal 
  to 
  

  

  /Ma 
  

  

  Calling 
  this 
  ts^ 
  we 
  have 
  

  

  3E 
  

  

  and 
  

  

  sinfl, 
  1 
  a 
  ; 
  

  

  —~ 
  = 
  1—^6 
  1 
  approximately, 
  

  

  = 
  1- 
  

  

  '2 
  & 
  ^ 
  

  

  Now 
  any 
  chosen 
  term 
  of 
  (1) 
  is 
  a 
  particular 
  solution 
  of 
  the 
  problem 
  ; 
  

   that 
  is 
  to 
  say, 
  it 
  is 
  the 
  solution 
  for 
  the 
  case 
  for 
  which 
  the 
  initial 
  distribu- 
  

   tion 
  of 
  temperature 
  is 
  that 
  which 
  it 
  expresses 
  when 
  £=0. 
  Hence 
  

  

  / 
  lEa^ 
  2 
  \ 
  

  

  e 
  ac 
  

  

  