﻿Penetrating 
  Radiation 
  from 
  the 
  Earth. 
  605 
  

  

  The 
  analysis 
  was 
  extended 
  by 
  the 
  writer 
  * 
  and 
  the 
  gradient 
  

   expressed 
  in 
  terms 
  of 
  a 
  transcendental 
  function 
  denoted 
  by 
  

  

  /(aO 
  ==«--*+*£*(-*), 
  (1) 
  

  

  E?( 
  — 
  x) 
  representing 
  Grlaisher's 
  exponential 
  integral. 
  The 
  

   function 
  /( 
  x) 
  is 
  of 
  special 
  importance 
  in 
  all 
  problems 
  relat- 
  

   ing 
  to 
  radiation 
  from 
  plates 
  which 
  radiate 
  from 
  each 
  element 
  

   of 
  volume 
  and 
  also 
  absorb 
  their 
  own 
  radiations 
  exponen- 
  

   tially 
  f. 
  

  

  The 
  object 
  of 
  the 
  present 
  note, 
  undertaken 
  at 
  the 
  sugges- 
  

   tion 
  of 
  Professor 
  A. 
  S. 
  Eve, 
  is 
  to 
  set 
  forth 
  the 
  formulas 
  

   for 
  the 
  gradient 
  of 
  the 
  penetrating 
  radiation 
  and 
  interpret 
  

   them 
  numerically 
  in 
  terms 
  of 
  the 
  constants 
  which 
  have 
  been 
  

   determined 
  since 
  the 
  papers 
  above 
  referred 
  to 
  were 
  published. 
  

   A 
  brief 
  account 
  of 
  existing 
  observations 
  bearing 
  on 
  the 
  

   subject 
  is 
  also 
  given 
  and 
  the 
  results 
  discussed. 
  

  

  Sect. 
  2. 
  Calculation 
  of 
  the 
  Gradients 
  of 
  the 
  Penetrating 
  

   Radiation. 
  

  

  The 
  following 
  notation 
  is 
  employed 
  in 
  the 
  sequel 
  : 
  — 
  

  

  n 
  x 
  — 
  number 
  of 
  ions 
  produced 
  per 
  second 
  per 
  c.c. 
  at 
  a 
  

   height 
  z 
  1 
  above 
  the 
  earth's 
  surface. 
  

  

  n 
  2 
  refers 
  to 
  the 
  number 
  of 
  ions 
  produced 
  per 
  c.c. 
  per 
  sec. 
  

   in 
  a 
  small 
  cavity 
  at 
  a 
  depth 
  z 
  2 
  below 
  the 
  earth's 
  surface. 
  

  

  pi 
  and 
  fjb 
  2 
  are 
  the 
  mean 
  coefficients 
  of 
  absorption 
  of 
  7-rays 
  

   by 
  air 
  at 
  ordinary 
  pressure 
  and 
  temperature 
  of 
  rock 
  respec- 
  

   tively. 
  

  

  For 
  air 
  Chadwick 
  { 
  obtains 
  the 
  value 
  /^ 
  1 
  /D 
  l 
  = 
  , 
  048, 
  Dj 
  

   being 
  the 
  density 
  of 
  the 
  air 
  : 
  this 
  gives 
  yLt 
  1 
  = 
  6'0 
  x 
  10" 
  era*" 
  , 
  

   considerably 
  higher 
  than 
  the 
  value 
  previously 
  obtained 
  by 
  

   Eve§andHess|| 
  (^ 
  = 
  4-4 
  x 
  10 
  " 
  5 
  cm." 
  'and 
  4*47 
  x 
  10 
  " 
  5 
  cm." 
  \ 
  

   respectively). 
  

  

  For 
  rock 
  Rutherford 
  IF 
  takes 
  the 
  value 
  ^ 
  2 
  /D 
  2 
  = 
  *04, 
  not 
  

   very 
  different 
  from 
  the 
  value 
  obtained 
  by 
  McClelland 
  

   (/z, 
  2 
  /D 
  2 
  = 
  , 
  034). 
  Taking 
  as 
  a 
  rough 
  value 
  D 
  2 
  = 
  3*0, 
  we 
  have 
  

   ^ 
  2 
  = 
  *12cm. 
  _1 
  . 
  

  

  We 
  denote 
  by 
  n 
  Q 
  the 
  number 
  of 
  ions 
  produced 
  per 
  c.c. 
  per 
  

   second 
  in 
  air 
  at 
  normal 
  temperature 
  and 
  pressure 
  at 
  a 
  distance 
  

  

  * 
  King, 
  Phil. 
  Mag. 
  xxiii. 
  Feb. 
  1912, 
  p. 
  242. 
  

  

  t 
  Applications 
  to 
  radiation 
  problems 
  are 
  given 
  by 
  the 
  writer, 
  King, 
  

   Phil. 
  Trans. 
  495 
  A, 
  vol. 
  ccxii. 
  p. 
  375 
  (1913). 
  

  

  X 
  Chadwick, 
  Proc. 
  Lond. 
  Phys. 
  Soc. 
  xxiv. 
  p. 
  153 
  (1912). 
  

  

  § 
  Eve, 
  Phil. 
  Mag. 
  July 
  1911, 
  p. 
  12. 
  

  

  |l 
  Hess, 
  Phys. 
  Zeit. 
  xii. 
  Nov. 
  1911, 
  p. 
  998. 
  

  

  IT 
  Rutherford, 
  * 
  Radio-activity,' 
  1913, 
  p. 
  637. 
  

  

  