﻿Forces 
  produced 
  by 
  Flowing 
  Solutions. 
  587 
  

  

  forced 
  through 
  glass 
  tubes, 
  have 
  formed 
  the 
  subject 
  of 
  the 
  

   theory 
  of 
  Hehnholtz*. 
  This 
  is 
  based 
  on 
  the 
  facts 
  that, 
  

   firstly, 
  when 
  a 
  conducting 
  liquid 
  streams 
  through 
  a 
  channel 
  

   of 
  insulating 
  material, 
  an 
  electromotive 
  force 
  is 
  produced 
  in 
  

   the 
  direction 
  of 
  the 
  liquid 
  stream 
  or 
  opposite 
  to 
  it, 
  while, 
  

   secondly, 
  an 
  electromotive 
  force 
  applied 
  at 
  the 
  ends 
  of 
  the 
  

   channel, 
  or 
  between 
  two 
  points 
  in 
  it 
  separated 
  by 
  a 
  convenient 
  

   distance, 
  produces 
  a 
  flow 
  of 
  liquid. 
  Helmholtz 
  assumed 
  that 
  

   there 
  existed 
  a 
  difference 
  of 
  potential 
  at 
  the 
  boundary 
  between 
  

   the 
  insulating 
  material 
  of 
  the 
  tube 
  and 
  the 
  conducting 
  liquid, 
  

   and 
  derived 
  an 
  equation 
  connecting 
  this 
  difference 
  of 
  potential 
  

   </>i 
  — 
  <^a 
  (where 
  <pi 
  is 
  the 
  potential 
  of 
  the 
  liquid, 
  <^„ 
  that 
  of 
  the 
  

   solid 
  insulator) 
  with 
  the 
  pressure 
  P 
  between 
  two 
  cross- 
  

   sections 
  of 
  the 
  channel^ 
  the 
  electromotive 
  force 
  E 
  existing 
  

   between 
  these 
  two 
  cross-sections, 
  the 
  specific 
  conductivity 
  /c 
  

   of 
  the 
  solution, 
  and 
  its 
  coefficient 
  of 
  internal 
  friction 
  t] 
  : 
  

  

  The 
  formula 
  applies 
  to 
  tubes 
  for 
  which 
  Poiseuille''s 
  law 
  

   holds, 
  provided 
  also 
  the 
  potential 
  difference 
  between 
  the 
  two 
  

   cross-sections 
  is 
  not 
  led 
  into 
  an 
  outside 
  circuit, 
  but 
  only 
  

   produces 
  an 
  electric 
  current 
  along 
  the 
  tube, 
  and 
  that 
  there 
  

   is 
  no 
  shifting 
  of 
  the 
  last 
  layer 
  of 
  the 
  liquid 
  adhering 
  to 
  the 
  

   glass. 
  This 
  last 
  condition 
  may 
  be 
  expressed 
  in 
  the 
  form 
  that 
  

   the 
  external 
  friction 
  is 
  infinite 
  ; 
  for 
  a 
  finite 
  value 
  the 
  term 
  

   <^i 
  — 
  ^a 
  has 
  to 
  be 
  replaced 
  by 
  

  

  <^.-*„+?||, 
  

  

  where 
  ^ 
  is 
  the 
  constant 
  of 
  slipping, 
  and 
  N 
  the 
  normal 
  to 
  the 
  

   tube 
  wall. 
  To 
  this, 
  the 
  " 
  electric 
  moment 
  of 
  the 
  double 
  

   layer," 
  Dorn 
  gave 
  the 
  symbol 
  //,. 
  Lambj 
  has 
  proposed 
  a 
  

   modification 
  of 
  the 
  Helmholtz 
  theory, 
  which 
  differs 
  only 
  by 
  

   the 
  assumption 
  that 
  the 
  charge 
  of 
  the 
  liquid 
  is 
  rigidly 
  con- 
  

   fined 
  to 
  the 
  last 
  layer. 
  In 
  Lamb's 
  theory 
  the 
  same 
  term 
  

  

  takes 
  the 
  form 
  (</>* 
  — 
  ^a);,, 
  where 
  d 
  is 
  the 
  thickness 
  of 
  the 
  

  

  double 
  layer, 
  i. 
  e. 
  a 
  magnitude 
  of 
  the 
  order 
  10~^, 
  while 
  I 
  is 
  the 
  

   ratio 
  of 
  internal 
  and 
  external 
  friction, 
  i. 
  e. 
  of 
  the 
  coefficients 
  

   of 
  friction 
  and 
  slip. 
  

  

  Helmholtz 
  verified 
  his 
  theory 
  by 
  applying 
  to 
  it 
  the 
  experi- 
  

   mental 
  results 
  of 
  Quincke 
  t, 
  who 
  forced 
  water 
  through 
  two 
  

  

  * 
  Wied. 
  Ann. 
  vii. 
  p. 
  337 
  (1879). 
  

   t 
  Phil. 
  Mag. 
  [5] 
  XXV. 
  p. 
  52 
  (1898). 
  

   \ 
  Pogg. 
  Ann. 
  cvii. 
  p. 
  1 
  (1859; 
  ; 
  cviii. 
  p. 
  33 
  (1860). 
  

   2R2 
  

  

  