﻿Bessel 
  Functions 
  of 
  Equal 
  Argument 
  and 
  Order. 
  271 
  

   of 
  the 
  water 
  in 
  the 
  cell 
  was 
  also 
  determined 
  by 
  polarization. 
  

   I£ 
  - 
  - 
  is 
  a 
  small 
  quantity 
  and 
  G 
  is 
  the 
  original 
  concen- 
  

   tration, 
  the 
  actual 
  concentration 
  will 
  be 
  

  

  a 
  

  

  When 
  the 
  compressor 
  is 
  used 
  we 
  shall 
  generally 
  find 
  a 
  = 
  a, 
  

   and 
  in 
  most 
  cases 
  g 
  has 
  been 
  so 
  small 
  as 
  not 
  to 
  be 
  detectable 
  

   by 
  polarization. 
  

  

  The 
  angle 
  of 
  polarization 
  is 
  measured 
  by 
  Laurent's 
  pola- 
  

   rimeter, 
  which 
  enabled 
  me 
  to 
  read 
  with 
  an 
  accuracy 
  of 
  1 
  or 
  2 
  

   minutes, 
  when 
  taking 
  the 
  mean 
  of 
  a 
  number 
  of 
  readings. 
  

   With 
  the 
  tube 
  used 
  it 
  corresponds 
  to 
  a 
  concentration 
  of 
  

   about 
  1/8 
  — 
  1/4 
  gr. 
  in 
  the 
  litre. 
  

  

  [To 
  be 
  continued."] 
  

  

  XX. 
  On 
  Bessel 
  Functions 
  of 
  Equal 
  Argument 
  and 
  Order. 
  

   By 
  J. 
  W. 
  Nicholson, 
  D.Sc, 
  B.A., 
  Isaac 
  Newton 
  Student 
  

   in 
  the 
  University 
  of 
  Cambridge*. 
  

  

  npHE 
  solution 
  of 
  many 
  physical 
  problems 
  depends 
  upon 
  a 
  

   JL 
  knowledge 
  of 
  the 
  behaviour 
  of 
  the 
  Bessel 
  function 
  

   commonly 
  denoted 
  by 
  J 
  n 
  (z), 
  and 
  of 
  other 
  functions 
  associated 
  

   with 
  it, 
  when 
  the 
  argument 
  z 
  and 
  order 
  n 
  are 
  nearly 
  

   equal. 
  The 
  only 
  treatment 
  of 
  this 
  question 
  which 
  the 
  author 
  

   has 
  been 
  able 
  to 
  discover 
  is 
  contained 
  in 
  a 
  paper 
  by 
  Lorenz 
  f, 
  

   which 
  only 
  deals 
  with 
  a 
  very 
  special 
  case 
  in 
  which 
  n 
  is 
  half 
  

   of 
  an 
  odd 
  integer. 
  But 
  even 
  in 
  this 
  restricted 
  investigation, 
  

   the 
  method 
  employed 
  is 
  highly 
  unsatisfactory, 
  and 
  many 
  of 
  

   the 
  steps 
  made 
  seem 
  incapable 
  of 
  justification. 
  For 
  example, 
  

   at 
  one 
  point 
  Lorenz 
  divides 
  a 
  definite 
  integral, 
  involving 
  a 
  

   Bessel 
  function 
  in 
  the 
  integrand, 
  into 
  two 
  parts, 
  say 
  a 
  

   and 
  ft. 
  In 
  a 
  the 
  range 
  extends 
  from 
  zero 
  to 
  a 
  " 
  small 
  

   quantity 
  7i,'' 
  and 
  it 
  is 
  shown 
  that 
  to 
  a 
  certain 
  order 
  a 
  may 
  

   be 
  neglected. 
  The 
  range 
  in 
  ft 
  extends 
  from 
  h 
  to 
  a 
  quantity 
  

  

  " 
  not 
  small," 
  in 
  fact 
  — 
  . 
  Lorenz 
  substitutes 
  an 
  asymptotic 
  

  

  expansion 
  for 
  the 
  Bessel 
  function 
  of 
  zero 
  order 
  in 
  ft, 
  but 
  

   thereby 
  renders 
  the 
  result 
  liable 
  to 
  an 
  error 
  of 
  the 
  same 
  

   magnitude 
  as 
  the 
  terms 
  retained, 
  for 
  in 
  the 
  lower 
  portion 
  of 
  

  

  * 
  Communicated 
  by 
  the 
  Author. 
  

  

  t 
  " 
  Sur 
  la 
  reflexion 
  de 
  la 
  lumiere 
  par 
  une 
  sphere 
  transparente," 
  G£uvres 
  

   Scientifiques, 
  i. 
  pp. 
  435 
  et 
  seq. 
  

  

  