﻿Multiple 
  Atojnic 
  Disintegration. 
  743 
  

  

  or 
  end-product 
  initially 
  ; 
  T, 
  U, 
  and 
  E, 
  the 
  existing 
  quantities 
  of 
  

   thorium, 
  uranium, 
  and 
  end-product 
  of 
  radium 
  ; 
  Xa, 
  \x 
  the 
  con- 
  

   stants 
  of 
  the 
  thorium 
  and 
  radium 
  modes 
  in 
  the 
  multiple 
  dis- 
  

   integration 
  of 
  uranium 
  ; 
  Xb 
  the 
  constant 
  .of 
  thorium 
  itself, 
  and 
  

   k 
  is 
  the 
  fraction 
  of 
  the 
  number 
  of 
  uranium 
  atoms 
  undergoing 
  

   the 
  radium 
  mode 
  of 
  disintegration 
  \Nhich 
  eventuaHv 
  become 
  

   atoms 
  of 
  the 
  end-product 
  in 
  question. 
  Were 
  all 
  subsequent 
  

   disintegrations 
  simple 
  h 
  of 
  course 
  would 
  be 
  unity. 
  The 
  effect 
  

   of 
  the 
  intermediate 
  bodies 
  between 
  uranium 
  and 
  the 
  end- 
  

   product 
  has 
  been 
  neglected. 
  So 
  far 
  as 
  is 
  known 
  this 
  would 
  

   be 
  small. 
  Unfortunately 
  on 
  account 
  of 
  the 
  long 
  period 
  of 
  

   thorium 
  and 
  the 
  isomorphism 
  of 
  uranium 
  and 
  thorium 
  oxides 
  

   it 
  is 
  doubtful 
  if 
  much 
  evidence 
  can 
  be 
  derived 
  from 
  the 
  

   composition 
  of 
  minerals. 
  Thorianite 
  is 
  regarded 
  as 
  an 
  

   extremely 
  old 
  mineral 
  geologically 
  yet 
  it 
  contains 
  so 
  great 
  a 
  

   percentage 
  of 
  thorium 
  that 
  from 
  the 
  point 
  of 
  view 
  of 
  radio- 
  

   activity 
  it 
  must 
  be 
  a 
  recent 
  formation. 
  As 
  Boltwood 
  has 
  

   pointed 
  out 
  Qoc. 
  cit. 
  p. 
  262), 
  Hillebrand's 
  analyses 
  of 
  

   uranium 
  minerals 
  show 
  generally 
  that 
  the 
  percentage 
  of 
  

   total 
  rare 
  earths 
  (less 
  perfectly 
  of 
  the 
  thorium) 
  increases 
  as 
  the 
  

   lead 
  increases 
  but 
  there 
  are 
  numerous 
  exceptions. 
  It 
  is 
  

   probable 
  that 
  geological 
  time 
  is 
  all 
  too 
  short 
  to 
  enable 
  the 
  

   necessary 
  evidence 
  to 
  be 
  obtained 
  from 
  the 
  composition 
  of 
  

   minerals. 
  

  

  Before 
  concluding 
  a 
  special 
  case 
  may 
  be 
  referred 
  to 
  which 
  

   has 
  some 
  points 
  of 
  interest 
  and 
  might 
  very 
  probably 
  occur. 
  

   If 
  one 
  of 
  the 
  modes 
  of 
  disintegration 
  were 
  a 
  relatively 
  mild 
  

   form 
  of 
  rearrangement, 
  rather 
  than 
  an 
  explosive 
  disintegration, 
  

   it 
  is 
  conceivable 
  that 
  the 
  ray-producing 
  change 
  might 
  occur 
  

   with 
  the 
  same 
  period 
  independently 
  of 
  whether 
  the 
  rayless 
  

   change 
  had 
  or 
  had 
  not 
  occurred, 
  expelling 
  the 
  same 
  character 
  of 
  

   ray 
  in 
  each 
  case. 
  On 
  this 
  view 
  actinium 
  would 
  proceed 
  from 
  re- 
  

   arranged 
  but 
  undisintegrated 
  uranium 
  X, 
  radium 
  from 
  uranium 
  

   X 
  disintegrating 
  directly, 
  or 
  vice 
  versa. 
  If 
  X^ 
  represents 
  the 
  

   constant 
  of 
  the 
  rayless 
  and 
  Xx 
  that 
  of 
  the 
  ray-producing 
  mode 
  

   of 
  disintegration, 
  it 
  can 
  be 
  shown 
  for 
  this 
  case 
  that 
  the 
  rays 
  

   will 
  now 
  follow 
  the 
  exponential 
  changes 
  with 
  their 
  own 
  true 
  

   period, 
  Xx, 
  rather 
  than 
  with 
  the 
  X.^i-Ax 
  period. 
  This 
  holds 
  

   true 
  whether 
  the 
  rearranged 
  uranium 
  X 
  is 
  present 
  or 
  absent 
  

   initially. 
  The 
  calculation 
  which 
  gives 
  this 
  simple 
  result 
  is 
  

   lengthy 
  and 
  may 
  be 
  omitted. 
  If 
  7-rays 
  accompanied 
  the 
  

   /5-rays 
  in 
  the 
  direct 
  mode 
  of 
  change 
  of 
  the 
  uranium 
  X, 
  and 
  

   not 
  in 
  the 
  change 
  of 
  rearranged 
  uranium 
  X, 
  the 
  7-rays 
  would 
  

   decay 
  with 
  the 
  Xa 
  + 
  Xx 
  period, 
  the 
  /3-rays 
  according 
  to 
  the 
  

   Xx 
  period. 
  The 
  difference 
  in 
  period 
  would 
  be 
  small 
  and 
  

   might 
  well 
  be 
  overlooked. 
  

  

  