﻿8(5 
  Prof. 
  C. 
  Barus 
  on 
  the 
  JBehtiviour 
  of 
  the 
  

  

  detailed 
  in 
  the 
  above 
  paragraphs. 
  As 
  before, 
  let 
  n 
  be 
  the 
  

   number 
  of 
  particles 
  per 
  cubic 
  centim., 
  so 
  that 
  n 
  is 
  the 
  concen- 
  

   tration 
  or 
  density 
  of 
  distribution 
  o£ 
  the 
  phosphorus 
  emana- 
  

   tion. 
  Let 
  k 
  be 
  the 
  "absorption" 
  velocity 
  of 
  the 
  ion, 
  treated 
  

   in 
  the 
  first 
  instance 
  as 
  independent 
  of 
  the 
  potential 
  and 
  of 
  

   the 
  concentration 
  gradients. 
  Let 
  k' 
  be 
  the 
  coefficient 
  of 
  

   decay, 
  so 
  that 
  k'n 
  2 
  is 
  the 
  number 
  of 
  ions 
  vanishing 
  per 
  cubic 
  

   centim. 
  per 
  second. 
  Finally, 
  let 
  R 
  be 
  the 
  external 
  radius 
  of 
  

   the 
  condenser 
  and 
  C 
  its 
  effective 
  capacity 
  including 
  that 
  of 
  

   the 
  electrometer. 
  

  

  With 
  regard 
  to 
  the 
  electrical 
  currents, 
  let 
  V 
  be 
  the 
  

   potential 
  at 
  a 
  distance 
  r 
  from 
  the 
  centre 
  of 
  the 
  condenser 
  

   whose 
  external 
  face 
  is 
  put 
  to 
  earth. 
  Let 
  U 
  be 
  the 
  aggregate 
  

   velocity 
  of 
  the 
  ions 
  in 
  the 
  unit 
  electric 
  field 
  and 
  e 
  the 
  charge 
  

   of 
  each. 
  

  

  In 
  all 
  cases 
  the 
  observations 
  are 
  made 
  when 
  the 
  flux 
  is 
  

   stationary, 
  so 
  that 
  dn/dt 
  = 
  throughout, 
  for 
  any 
  shell. 
  More- 
  

   over, 
  as 
  shown 
  elsewhere, 
  the 
  effect 
  of 
  a 
  potential 
  gradient 
  is 
  

   but 
  a 
  negligible 
  contribution 
  to 
  the 
  number 
  of 
  ions 
  which 
  

   are 
  absorbed 
  by 
  the 
  outer 
  surface 
  of 
  the 
  condenser. 
  

  

  To 
  begin 
  with 
  the 
  simplest 
  cases 
  : 
  — 
  If 
  the 
  motion 
  of 
  the 
  

   ion 
  is 
  entirely 
  independent 
  of 
  dV/dr 
  and 
  n, 
  the 
  accumulation 
  

   in 
  an 
  elementary 
  shell 
  at 
  a 
  distance 
  r 
  from 
  the 
  centre 
  will 
  be 
  

   4:7rkd(r 
  2 
  n)/dr 
  . 
  dr, 
  per 
  second; 
  the 
  decay 
  per 
  second, 
  k'n' 
  2 
  i7rr 
  2 
  dt\ 
  

   Hence 
  

  

  d(r 
  2 
  n)ldv=(k'lk)n 
  2 
  r 
  2 
  , 
  

   or 
  if 
  A 
  is 
  a 
  constant, 
  

  

  l/n=r((k'/k) 
  + 
  Ar). 
  

  

  In 
  the 
  absence 
  of 
  decay, 
  1/A 
  = 
  n7 
  >2 
  so 
  that 
  A 
  is 
  the 
  reciprocal 
  

   of 
  the 
  concentration, 
  n 
  l9 
  at 
  a 
  distance 
  1 
  from 
  the 
  centre. 
  

   If 
  conduction 
  were 
  prompted 
  solely 
  by 
  the 
  ions 
  which 
  reach 
  

   the 
  external 
  shell 
  kept 
  at 
  V=0, 
  since 
  the 
  charge 
  in 
  this 
  

   shell 
  is 
  per 
  centim. 
  

  

  edR/(R(k'/k) 
  + 
  AR) 
  

  

  and 
  its 
  time 
  of 
  discharge 
  dR/k, 
  

  

  Cd\/dt 
  = 
  ±7rkeB./((V/k) 
  + 
  AR). 
  

  

  In 
  the 
  absence 
  of 
  decay, 
  A' 
  = 
  0, 
  and 
  

  

  dV/dt^AwkenJC, 
  

  

  where 
  tti 
  as 
  stated 
  holds 
  for 
  r—1. 
  This 
  case, 
  in 
  which 
  

   dY/dt 
  = 
  ds/dt 
  = 
  const., 
  independent 
  of 
  the 
  radius 
  of 
  the 
  con- 
  

   denser 
  is 
  effectually 
  excluded 
  by 
  the 
  observations 
  given 
  in 
  

   fig. 
  5. 
  If 
  k! 
  is 
  not 
  zero, 
  

  

  dY/dt= 
  (±7rke/C)(l/{k'/kR 
  + 
  A)), 
  

  

  