﻿280 
  G. 
  F. 
  Becker 
  — 
  Computing 
  Diffusion. 
  

  

  Art. 
  XXV. 
  — 
  Note 
  on 
  Computing 
  Diffusion 
  • 
  by 
  Geo. 
  F. 
  

  

  Becker. 
  

  

  Diffusive 
  phenomena 
  have 
  become 
  of 
  great 
  importance 
  to 
  

   geologists. 
  As 
  every 
  one 
  knows, 
  Lord 
  Kelvin's 
  famous 
  inquiry 
  

   into 
  the 
  age 
  of 
  the 
  earth 
  is 
  based 
  on 
  the 
  diffusion 
  of 
  heat 
  in 
  a 
  

   cooling 
  globe 
  of 
  large 
  radius. 
  The 
  modern 
  theory 
  of 
  the 
  dif- 
  

   ferentiation 
  of 
  rock 
  magmas 
  also 
  is 
  founded 
  on 
  phenomena 
  of 
  

   molecular 
  flow, 
  the 
  least 
  complex 
  of 
  which 
  is 
  the 
  simplest 
  case 
  

   of 
  liquid 
  diffusion. 
  The 
  diffusion 
  of 
  motion 
  in 
  a 
  viscous 
  fluid 
  

   is 
  subject 
  to 
  the 
  same 
  laws 
  and 
  is 
  capable 
  of 
  geological 
  applica- 
  

   tions. 
  Possibly 
  even 
  the 
  diffusion 
  of 
  current 
  density 
  in 
  a 
  

   homogeneous 
  conductor 
  may 
  eventually 
  be 
  made 
  to 
  contribute 
  

   to 
  a 
  knowledge 
  of 
  the 
  earth's 
  interior. 
  

  

  The 
  computation 
  of 
  *the 
  simplest 
  diffusive 
  phenomena 
  has 
  a 
  

   formidable 
  appearance 
  to 
  most 
  geologists, 
  while 
  to 
  those 
  who 
  

   are 
  mathematically 
  inclined 
  the 
  reckoning 
  appears 
  inelegant 
  

   and 
  clumsy. 
  It 
  is 
  probable 
  that 
  every 
  one 
  who 
  has 
  actually 
  

   computed 
  diffusions 
  has 
  seen 
  how 
  the 
  subject 
  could 
  be 
  simpli- 
  

   fied, 
  but 
  each 
  appears 
  to 
  have 
  shrunk 
  from 
  the 
  trouble 
  of 
  set- 
  

   ting 
  the 
  matter 
  straight. 
  I 
  shall 
  attempt 
  to 
  make 
  the 
  subject 
  

   so 
  easy 
  that 
  no 
  geologist 
  will 
  hesitate 
  to 
  compute 
  any 
  case 
  

   which 
  may 
  help 
  him 
  to 
  frame 
  a 
  theory 
  or 
  to 
  test 
  an 
  hypothesis. 
  

  

  In 
  "linear" 
  motion 
  as 
  defined 
  by 
  Fourier 
  the 
  subject 
  of 
  

   motion, 
  or 
  the 
  "quality 
  " 
  as 
  Lord 
  Kelvin 
  calls 
  it, 
  varies 
  only 
  in 
  

   one 
  direction 
  ; 
  in 
  other 
  words, 
  it 
  remains 
  uniform 
  at 
  all 
  points 
  

   in 
  any 
  one 
  plane 
  at 
  right 
  angles 
  to 
  this 
  direction. 
  Qualities 
  in 
  

   this 
  sense 
  are 
  for 
  example 
  the 
  temperature 
  of 
  a 
  body, 
  the 
  

   amount 
  of 
  substance 
  in 
  solution 
  per 
  unit 
  volume, 
  the 
  velocity 
  

   of 
  a 
  viscous 
  fluid. 
  For 
  qualities 
  obeying 
  the 
  law 
  of 
  diffusion 
  

   the 
  differential 
  equation 
  is 
  

  

  dv 
  _ 
  d 
  2 
  v 
  

   dt 
  ~ 
  dx 
  2 
  ' 
  

  

  Here 
  v 
  is 
  the 
  quality, 
  t 
  the 
  time, 
  x 
  the 
  distance 
  from 
  the 
  plane 
  

   of 
  contact 
  between 
  the 
  subject 
  of 
  diffusion 
  and 
  the 
  medium 
  

   into 
  which 
  it 
  diffuses, 
  and 
  k 
  is 
  the 
  u 
  diffusivity 
  " 
  assumed 
  to 
  be 
  

   constant. 
  The 
  equation 
  may 
  be 
  expressed 
  by 
  the 
  statement 
  

   that 
  the 
  time 
  rate 
  of 
  change 
  of 
  quality 
  is 
  proportional 
  to 
  the 
  

   space 
  rate 
  of 
  the 
  space 
  rate 
  of 
  change 
  of 
  quality. 
  It 
  is 
  usual 
  

   with 
  English 
  writers 
  to 
  apply 
  the 
  C. 
  G. 
  S. 
  system 
  to 
  these 
  

   measurements. 
  That 
  system 
  attains 
  a 
  most 
  convenient 
  uni- 
  

   formity, 
  though 
  only 
  at 
  the 
  expense 
  of 
  some 
  very 
  awkward 
  

   numbers. 
  

  

  The 
  simplest 
  case 
  of 
  linear 
  diffusion 
  arises 
  when 
  the 
  diffus- 
  

   ing 
  quality 
  at 
  the 
  initial 
  plane 
  is 
  kept 
  constant 
  and 
  when 
  the 
  

  

  

  