﻿Relation 
  between 
  Uranhnn 
  and 
  Radium. 
  855 
  

  

  Solving 
  the 
  equations, 
  the 
  growth 
  of 
  radium 
  is 
  given 
  by 
  

  

  The 
  last 
  term 
  is 
  always 
  very 
  small, 
  and 
  may 
  be 
  neglected 
  

   at 
  once. 
  The 
  penultimate 
  term 
  approaches 
  to 
  zero 
  as 
  t 
  

   increases. 
  After 
  a 
  period 
  several 
  times 
  that 
  of 
  uranium 
  A 
  

   it 
  may 
  be 
  neglected. 
  The 
  expression 
  then 
  becomes 
  

  

  If 
  instead 
  of 
  reckoning 
  the 
  time 
  from 
  the 
  start 
  we 
  reckon 
  it 
  

   from 
  a 
  date 
  later 
  by 
  the 
  sum 
  of 
  the 
  periods 
  of 
  average 
  lite 
  of 
  

   the 
  two 
  short-lived 
  bodies, 
  that 
  is, 
  for 
  t 
  in 
  the 
  equation 
  

   we 
  substitute 
  T, 
  \\here 
  

  

  A2 
  A3 
  

  

  then 
  

   E 
  = 
  1/2 
  X1X4A 
  j 
  T^ 
  + 
  — 
  , 
  -f- 
  -L 
  I 
  =, 
  1/2 
  x{K,kT 
  + 
  constant. 
  

  

  That 
  is 
  to 
  say, 
  when 
  the 
  short-lived 
  intermediate 
  bodies 
  

   come 
  into 
  equilibrium 
  the 
  production 
  of 
  radium 
  proceeds 
  

   strictly 
  according 
  to 
  the 
  square 
  of 
  the 
  time 
  reckoned, 
  not 
  

   from 
  the 
  start, 
  but 
  from 
  a 
  date 
  later 
  by 
  the 
  sum 
  of 
  the 
  

   periods 
  of 
  the 
  intermediate 
  bodies. 
  

  

  To 
  show 
  the 
  efiect 
  of 
  the 
  intermediate 
  bodies 
  initially 
  the 
  

   gra{)hs 
  of 
  equation 
  (I.) 
  have 
  been 
  plotted 
  against 
  the 
  square 
  

   of 
  the 
  time, 
  for 
  the 
  following 
  values 
  of 
  X3 
  : 
  — 
  4, 
  2, 
  1, 
  '&&, 
  '5, 
  

   •33, 
  and 
  '25 
  (fig. 
  4, 
  p. 
  856). 
  The 
  curves 
  therefore 
  show 
  the 
  

   production 
  of 
  radium 
  according 
  as 
  the 
  hypothetical 
  body 
  

   uranium 
  A 
  has 
  the 
  period 
  of 
  three 
  months, 
  six 
  months, 
  

   1, 
  I'O, 
  2, 
  3, 
  and 
  4 
  years. 
  The 
  straight 
  line 
  uppermost 
  on 
  

   the 
  diagram 
  shows 
  the 
  production 
  of 
  radium 
  on 
  the 
  same 
  

   scale 
  if 
  no 
  short-lived 
  intermediate 
  products 
  intervened. 
  It 
  

   will 
  be 
  seen 
  at 
  once 
  from 
  these 
  curs'es 
  that 
  if 
  the 
  portion 
  

   between 
  the 
  third 
  and 
  fourth 
  years, 
  that 
  is 
  between 
  i^ 
  = 
  9 
  

   and 
  ^^=16, 
  is 
  examined, 
  even 
  for 
  the 
  lowest 
  curve 
  corre- 
  

   sponding 
  to 
  the 
  four-vear 
  period, 
  the 
  curve 
  departs 
  hut 
  little 
  

  

  3M2 
  

  

  