﻿364 
  Prof. 
  G. 
  N. 
  Watson 
  on 
  Bessel 
  Functions 
  

  

  of 
  the 
  cases 
  in 
  which 
  the 
  pair 
  is 
  observed, 
  while 
  they 
  act 
  as 
  

   combined 
  (exert 
  a 
  molecular 
  pressure) 
  in 
  the 
  remainder. 
  

   It 
  seems 
  to 
  me 
  that 
  it 
  is 
  a 
  theory 
  on 
  lines 
  similar 
  to 
  these 
  

   which 
  will 
  ultimately 
  succeed 
  in 
  reconciling 
  all 
  the 
  difficulties 
  

   connected 
  with 
  strong 
  electrolytes. 
  

  

  Summary. 
  

  

  (1) 
  A 
  critical 
  discussion 
  of 
  the 
  way 
  in 
  which 
  the 
  law 
  of 
  

   mass 
  action 
  fails 
  for 
  strong 
  electrolytes 
  leads 
  to 
  the 
  conclusion 
  

   that 
  the 
  reduction 
  in 
  the 
  molecular 
  conductivity 
  with 
  

   increasing 
  concentration 
  must 
  be 
  ascribed 
  mainly 
  to 
  a 
  

   reduction 
  in 
  the 
  mobilities 
  of 
  the 
  ions, 
  and 
  not 
  to 
  a 
  reduction 
  

   in 
  their 
  number 
  by 
  association 
  into 
  molecules. 
  

  

  (2) 
  A 
  theoretical 
  investigation 
  of 
  the 
  effect 
  of 
  interionic 
  

   force 
  shows 
  that 
  identical 
  variations 
  with 
  the 
  concentration 
  

   will 
  be 
  produced 
  in 
  the 
  conductivity 
  and 
  in 
  the 
  osmotic 
  

   pressure 
  of 
  the 
  "free" 
  ions 
  (as 
  denned 
  on 
  p. 
  357). 
  

  

  (3) 
  The 
  application 
  of 
  this 
  result 
  to 
  strong 
  electrolytes 
  

   shows 
  that 
  the 
  variation 
  in 
  the 
  conductivity 
  and 
  the 
  freezing- 
  

   point 
  can 
  be 
  best 
  explained 
  by 
  a 
  modification 
  in 
  the 
  view 
  we 
  

   take 
  of 
  what 
  constitutes 
  association. 
  According 
  to 
  this 
  ions 
  

   in 
  strong 
  electrolytes 
  are 
  not 
  associated 
  into 
  molecules 
  ; 
  they 
  

   are 
  neither 
  completely 
  associated 
  nor 
  completely 
  free, 
  but 
  

   pairs 
  of 
  ions 
  which 
  are 
  temporarily 
  nearest 
  together, 
  in 
  

   consequence 
  of 
  the 
  electric 
  forces 
  between 
  them, 
  will, 
  in 
  

   a 
  fraction 
  of 
  cases, 
  act 
  as 
  if 
  bound 
  together, 
  and 
  in 
  the 
  

   remaining 
  cases 
  as 
  if 
  free. 
  

  

  The 
  University, 
  Sheffield, 
  

   December 
  1917. 
  

  

  XL. 
  Bessel 
  Functions 
  of 
  Equal 
  Order 
  and 
  Argument. 
  By 
  

   G. 
  N. 
  Watson, 
  M.A., 
  D.Sc, 
  Assistant 
  Professor 
  of 
  Pure 
  

   Mathematics 
  at 
  University 
  College 
  , 
  London 
  *. 
  

  

  1. 
  A 
  PPROXIMATE 
  formulae 
  for 
  the 
  Bessel 
  function 
  

  

  J\. 
  and 
  its 
  derivate, 
  J 
  n 
  (n) 
  and 
  J„'(n), 
  (when 
  n 
  is 
  

  

  large) 
  have 
  been 
  discussed 
  in 
  numerous 
  papers 
  during 
  the 
  

  

  last 
  few 
  years 
  f 
  ; 
  several 
  of 
  these 
  papers 
  have 
  appeared 
  in 
  

  

  this 
  Magazine. 
  

  

  * 
  Communicated 
  by 
  the 
  Author. 
  

  

  t 
  Debye, 
  Math. 
  Ann. 
  lxvii. 
  pp. 
  535-558 
  (1909). 
  Rayleigh, 
  Phil. 
  

   Mag. 
  Dec. 
  1910. 
  Nicholson, 
  Phil. 
  Mag. 
  Dec. 
  1907, 
  Aug. 
  1908, 
  Feb. 
  

   1910. 
  Watson, 
  Proc. 
  London 
  Math. 
  Soc. 
  (2) 
  xvi. 
  pp. 
  150-174 
  (1917) 
  ; 
  

   Proc. 
  Camb. 
  Phil. 
  Soc. 
  xix. 
  pp. 
  42-48 
  (1917). 
  Various 
  numerical 
  results 
  

   have 
  also 
  been 
  given 
  by 
  Airey 
  in 
  a 
  series 
  of 
  recent 
  papers 
  in 
  the 
  Phil. 
  

   Mag. 
  

  

  