﻿Multiple 
  xitomic 
  Disintegration. 
  741 
  

  

  Q 
  of 
  the 
  substance 
  will 
  diminish 
  according 
  to 
  the 
  differential 
  

   equation 
  

  

  Qt 
  _ 
  -(Xa+.\x)< 
  

  

  where 
  Q; 
  is 
  the 
  quantity 
  remaining 
  at 
  time 
  t 
  and 
  Qq 
  the 
  

   initial 
  quantity. 
  

  

  That 
  is 
  to 
  say, 
  for 
  a 
  radio-element 
  undergoing 
  a 
  dual 
  

   disintegration 
  the 
  quantity 
  would 
  diminish 
  exponentially 
  

   with 
  the 
  time 
  as 
  in 
  a 
  simple 
  disintegration. 
  The 
  radioactive 
  

   constant 
  of 
  the 
  multiple 
  change 
  would 
  be 
  the 
  sum 
  of 
  the 
  

   separate 
  constants. 
  If, 
  for 
  example, 
  a-rays 
  were 
  given 
  out 
  

   in 
  one 
  of 
  the 
  modes 
  of 
  disintegration, 
  and 
  ^-rays 
  in 
  the 
  other, 
  

   each 
  type 
  of 
  radiation 
  would 
  decay 
  at 
  the 
  same 
  period, 
  namely, 
  

   the 
  sum 
  of 
  the 
  two 
  separate 
  periods. 
  The 
  same 
  holds 
  true 
  

   whatever 
  the 
  number 
  of 
  modes 
  of 
  disintegration. 
  So 
  that 
  

   multiple 
  disintegration 
  would 
  not 
  be 
  easy 
  to 
  detect 
  so 
  far 
  as 
  

   ordinary 
  criteria 
  go. 
  But 
  with 
  regard 
  to 
  the 
  products 
  

   formed 
  in 
  the 
  various 
  modes 
  there 
  will 
  be 
  constancy 
  of 
  pro- 
  

   portionality 
  between 
  them. 
  To 
  suggest 
  a 
  concrete 
  case, 
  let 
  

   ns 
  suppose 
  the 
  disintegration 
  of 
  uranium 
  X 
  is 
  dual, 
  one 
  mode 
  

   producing 
  ultimately 
  radium, 
  the 
  other 
  ultimately 
  actinium, 
  

   and 
  let 
  the 
  constant 
  X^ 
  ^ppty 
  fo 
  the 
  radium 
  mode 
  and 
  Xx 
  to 
  

   the 
  actinium 
  mode 
  of 
  disintegration. 
  Then 
  the 
  ratio 
  between 
  

   the 
  number 
  of 
  nranium 
  atoms 
  forming 
  radium 
  and 
  the 
  number 
  

  

  forming 
  actinium 
  is 
  given 
  by 
  r^. 
  If 
  the 
  further 
  disinte- 
  

  

  Xx 
  

  

  grations 
  of 
  both 
  these 
  elements 
  are 
  simple 
  the 
  same 
  ratio 
  

  

  will 
  express 
  the 
  relation 
  between 
  the 
  quantities 
  of 
  the 
  final 
  

  

  inactive 
  products 
  of 
  the 
  two 
  series. 
  This 
  suggests 
  at 
  once 
  

  

  a 
  method 
  of 
  obtaining 
  evidence 
  of 
  the 
  process 
  and 
  at 
  the 
  same 
  

  

  time 
  of 
  determining 
  the 
  nature 
  of 
  the 
  final 
  products. 
  If 
  a 
  

  

  constant 
  ratio 
  were 
  found 
  between 
  any 
  two 
  inactive 
  elements 
  

  

  present 
  in 
  uranium 
  minerals 
  it 
  would 
  be 
  very 
  strong 
  evidence 
  

  

  from 
  the 
  present 
  point 
  of 
  view. 
  

  

  According 
  to 
  the 
  researches 
  of 
  Bolt 
  wood 
  (Am. 
  Journ. 
  Sci. 
  

  

  1908, 
  p. 
  29G) 
  on 
  the 
  proportionate 
  a-raiiation 
  contributed 
  by 
  

  

  the 
  separate 
  constituents 
  in 
  equilibrium 
  in 
  uranium 
  minerals, 
  

  

  it 
  is 
  possible 
  to 
  deduce 
  (Phil. 
  Mag. 
  Oct. 
  1908, 
  p. 
  517) 
  

  

  that 
  of 
  eight 
  atoms 
  of 
  uranium 
  seven 
  go 
  to 
  form 
  radium 
  and 
  

  

  one 
  to 
  form 
  actinium. 
  The 
  ratio 
  -^ 
  is 
  therefore 
  equal 
  to 
  7. 
  

  

  