﻿742 
  Mr. 
  F. 
  Soddy 
  on 
  

  

  This 
  is 
  also 
  the 
  ratio 
  between 
  the 
  quantities 
  of 
  the 
  end- 
  

   products 
  of 
  the 
  two 
  series 
  in 
  minerals 
  on 
  the 
  assumption 
  stated. 
  

   , 
  Proceeding 
  with 
  the 
  idea 
  that 
  the 
  disintegration 
  of 
  

   uranium 
  X 
  is 
  dual, 
  it 
  makes 
  no 
  difference 
  whether 
  we 
  regard 
  

   the 
  l3' 
  and 
  7-rays 
  as 
  derived 
  from 
  one 
  only 
  or 
  from 
  both 
  of 
  

   the 
  modes 
  of 
  disintegration. 
  Or, 
  we 
  may 
  even 
  suppose 
  

   that 
  the 
  /8-rays 
  come 
  from 
  the 
  radium 
  mode, 
  while 
  7-rays 
  

   come 
  from 
  the 
  actinium 
  mode 
  of 
  disintegration, 
  which 
  appears 
  

   to 
  have 
  something 
  to 
  recommend 
  it 
  as 
  it 
  would 
  explain 
  the 
  

   relative 
  poverty 
  of 
  uranium 
  in 
  7-rays 
  (Soddy 
  and 
  Russell^ 
  

   Phil. 
  Mag. 
  Oct. 
  1909, 
  p. 
  631). 
  Both 
  types 
  of 
  rays 
  would 
  

   decay 
  at 
  the 
  same 
  rate. 
  The 
  apparent 
  value 
  of 
  the 
  radioactive 
  

   constant 
  of 
  uranium 
  X 
  (about 
  '031 
  (day)"^ 
  ) 
  would 
  be 
  the 
  

   sum 
  of 
  the 
  two 
  separate 
  constants, 
  one 
  of 
  which 
  is 
  seven 
  

   times 
  the 
  other. 
  The 
  radium 
  mode 
  would 
  thus 
  have 
  the 
  

   constant 
  '027 
  and 
  the 
  actinium 
  mode 
  of 
  disintegration, 
  '004. 
  

   This 
  point 
  of 
  view 
  explains 
  perfectly 
  the 
  known 
  relations 
  

   between 
  uranium, 
  actinium, 
  and 
  radium, 
  preserving 
  the^ 
  

   inevitableness 
  of 
  the 
  phenomenon, 
  without 
  the 
  necessity 
  of 
  

   supposing, 
  as 
  on 
  Rutherford's 
  theory 
  of 
  a 
  side-chain, 
  that 
  

   actinium 
  is 
  not 
  a 
  lineal 
  descendant 
  of 
  uranium 
  in 
  the 
  same 
  

   sense 
  as 
  radium 
  is 
  (' 
  Radioactive 
  Transformations,' 
  p. 
  177). 
  

   Considering 
  the 
  long 
  series 
  of 
  successive 
  disintegrations 
  

   passed 
  through 
  by 
  the 
  radioactive 
  atoms 
  it 
  would 
  be 
  rather 
  re- 
  

   markable 
  from 
  one 
  point 
  of 
  view 
  if 
  multiple 
  disintegration 
  

   did 
  not 
  sometimes 
  occur. 
  The 
  relations 
  between 
  thorium 
  

   and 
  uranium 
  in 
  minerals 
  would 
  receive 
  perhaps 
  a 
  better 
  

   explanation 
  on 
  the 
  view 
  that 
  thorium 
  is 
  one 
  product 
  of 
  a 
  

   multiple 
  disintegration 
  in 
  the 
  uranium 
  series. 
  The 
  evidence 
  

   has 
  been 
  discussed 
  by 
  Boltwood 
  (Am. 
  Jour. 
  Sci., 
  ] 
  905, 
  xx. 
  

   p. 
  256) 
  who 
  concluded 
  that 
  it 
  was 
  most 
  probable 
  that 
  thorium 
  

   was 
  a 
  disintegration 
  product 
  of 
  uranium. 
  The 
  period 
  of 
  

   thorium 
  is 
  probably 
  five 
  times 
  as 
  long 
  as 
  that 
  of 
  uranium,, 
  

   and 
  hence 
  thorium 
  cannot 
  be 
  intermediate 
  between 
  uranium 
  

   and 
  radium 
  as 
  in 
  this 
  case 
  there 
  should 
  be 
  a 
  constant 
  ratio 
  

   between 
  thorium 
  and 
  radium 
  in 
  equilibrium 
  and 
  not 
  between 
  

   uranium 
  and 
  radium. 
  On 
  the 
  present 
  idea 
  the 
  ratio 
  of 
  

   thorium 
  to 
  uranium 
  and 
  to 
  the 
  end-products 
  of 
  radium 
  can 
  

   be 
  calculated 
  but 
  the 
  expressions 
  involve 
  the 
  age 
  of 
  the 
  

   mineral. 
  These 
  expressions 
  are 
  

  

  ^ 
  = 
  >-A 
  Q__ 
  (\a+Xx-Ab)^ 
  

  

  U 
  X^-X^-Xx^ 
  ^ 
  ^ 
  

  

  T 
  _ 
  X.a(X 
  a 
  + 
  Xx) 
  /e-^■^^-€-(^^+^^)^ 
  

   E"" 
  A-Xx(^a 
  + 
  Xx-Xb) 
  V 
  i_e-'^^+^^^^ 
  / 
  

   where 
  t 
  is 
  the 
  age 
  of 
  a 
  mineral, 
  assumed 
  to 
  contain 
  no 
  thorium 
  

  

  