﻿Osmosis 
  and 
  Osmotic 
  Pressure. 
  

  

  267 
  

  

  And 
  i£ 
  the 
  function 
  yfr 
  really 
  varies 
  with 
  the 
  leaks, 
  the 
  

   solution 
  of 
  equations 
  3a 
  and 
  3b 
  must 
  give 
  different 
  pressures. 
  

  

  As 
  earlier 
  mentioned, 
  as 
  long 
  as 
  in 
  the 
  functions 
  ^r 
  the 
  

   leaks 
  are 
  different 
  from 
  zero, 
  the 
  quantities 
  dependent 
  on 
  

   the 
  membrane 
  will 
  not 
  disappear 
  and 
  the 
  pressure 
  7r 
  ' 
  will 
  

   be 
  a 
  function 
  of 
  the 
  membrane. 
  7r 
  / 
  is 
  the 
  pressure 
  we 
  should 
  

   get 
  if 
  the 
  total 
  leak 
  could 
  be 
  regarded 
  as 
  a 
  solution 
  leak 
  not 
  

   influencing 
  the 
  osmotic 
  flow. 
  This 
  is 
  the 
  pressure 
  which 
  is 
  

   measured 
  by 
  Berkeley 
  and 
  Hartley. 
  

  

  We 
  see 
  from 
  this 
  that 
  if 
  some 
  of 
  the 
  quantities 
  Z 
  x 
  l 
  2 
  ... 
  l 
  n 
  

   are 
  somewhat 
  great, 
  it 
  may 
  seem 
  as 
  if 
  the 
  pressure 
  ttq 
  is 
  a 
  

   property 
  mainly 
  due 
  to 
  the 
  membrane. 
  

  

  We 
  shall 
  find 
  the 
  mathematical 
  expression 
  for 
  the 
  ideal 
  

   osmotic 
  pressure 
  in 
  the 
  case 
  that 
  the 
  leaks 
  are 
  very 
  small 
  

   quantities. 
  We 
  have 
  earlier 
  assumed 
  that 
  the 
  function 
  yjr 
  is 
  

   continuous 
  in 
  the 
  neighbourhood 
  of 
  the 
  point 
  1 
  = 
  0. 
  We 
  

   will 
  now 
  also 
  assume 
  that 
  the 
  first 
  derivates 
  with 
  regard 
  to 
  

   the 
  pressure 
  and 
  the 
  leaks 
  are 
  continuous. 
  

  

  Then 
  we 
  can 
  apply 
  to 
  the 
  equation 
  36 
  the 
  formula 
  of 
  

   Taylor 
  for 
  the 
  neighbourhood 
  of 
  the 
  point 
  7r 
  = 
  7r 
  and 
  Z 
  = 
  0, 
  

   and 
  we 
  get 
  

  

  o=fOo'z,',y 
  ...C)=f 
  Or„oo 
  ... 
  0) 
  +(|J) 
  (»,'-«■; 
  

  

  + 
  

  

  t)M 
  

  

  

  V+ 
  

  

  (i?)/- 
  ,+ 
  - 
  

  

  But 
  according 
  to 
  the 
  definition 
  of 
  7r 
  

  

  ^(tt 
  ...0)=0. 
  

  

  If 
  =^l 
  i 
  s 
  a 
  finite 
  quantity 
  different 
  from 
  zero, 
  we 
  see 
  that 
  

  

  7r 
  — 
  ttq 
  will 
  be 
  a 
  quantity 
  of 
  the 
  same 
  order 
  as 
  the 
  leaks 
  

   and 
  will 
  disappear 
  with 
  the 
  leaks. 
  w 
  2 
  consists 
  of 
  terms 
  of 
  

   second 
  and 
  higher 
  order. 
  If 
  we 
  restrict 
  ourselves 
  to 
  terms 
  

   of 
  the 
  first 
  order 
  we 
  p'et 
  

  

  _i_ 
  n-b±\ 
  

  

  VdWo 
  

  

  + 
  ti" 
  

  

  k' 
  + 
  

  

  

  •'} 
  

  

  (4) 
  

  

  The 
  pressure 
  tt 
  ' 
  is 
  defined 
  by 
  the 
  equation 
  \ 
  = 
  0. 
  As 
  we 
  

   have 
  seen 
  we 
  can 
  put 
  

  

  x 
  = 
  SO), 
  

  

  l 
  = 
  V 
  (tt), 
  

  

  X' 
  = 
  f(,r). 
  

  

  T2 
  

  

  