﻿Dr. 
  J. 
  W. 
  Nicholson 
  on 
  the 
  Relation 
  of 
  

  

  As 
  Kelvin 
  has 
  shown 
  *, 
  the 
  first 
  integral 
  of 
  (15) 
  takes 
  

   approximately 
  the 
  form 
  

  

  __ 
  co 
  s(o-o^-Av'g--j:7r) 
  ..^ 
  

  

  ^ 
  ~ 
  ^{-2nTtd''aldh')' 
  • 
  • 
  • 
  ^ 
  ^ 
  

  

  while 
  the 
  second 
  integral 
  is 
  relatively 
  negligible. 
  For 
  

   example 
  in 
  the 
  case 
  of 
  deep 
  water 
  waves, 
  where 
  cr 
  = 
  \/{gk), 
  

   (18) 
  gives 
  

  

  ko 
  = 
  gty4:a\ 
  (To 
  = 
  gt/2.v, 
  

   so 
  that 
  

  

  o-Qt 
  - 
  JcqX 
  — 
  l7r 
  = 
  c/t^l4:,v 
  — 
  i 
  TT 
  ; 
  

   and 
  finally 
  

  

  It' 
  we 
  attempt 
  to 
  fill 
  up 
  the 
  gaps 
  in 
  our 
  solution 
  by 
  

   apptying 
  quadratures 
  to 
  (2), 
  we 
  have 
  to 
  face 
  the 
  difficulty 
  

   that, 
  as 
  written, 
  the 
  integral 
  is 
  not 
  convergent. 
  Some 
  

   analytical 
  transformation 
  is 
  called 
  for. 
  One 
  way 
  out 
  of 
  

   the 
  difficulty 
  might 
  be 
  to 
  calculate 
  the 
  difh^ence 
  between 
  the 
  

   solutions 
  for 
  finite 
  and 
  infinite 
  depth, 
  for 
  it 
  would 
  appear 
  

   that 
  the 
  non-convergent 
  part 
  of 
  the 
  integral, 
  corresponding 
  

   to 
  infinitely 
  small 
  wave- 
  lengths, 
  must 
  be 
  the 
  same 
  for 
  both. 
  

  

  Terling 
  Place, 
  Witham, 
  

   Maj 
  4, 
  1909. 
  

  

  11. 
  On 
  the 
  Relation 
  of 
  Airy^s 
  Integral 
  to 
  the 
  Bessel 
  Functions. 
  

   By 
  J. 
  W. 
  Nicholson, 
  M.A^ 
  1).Sc., 
  Isaac 
  Neicton 
  Student 
  

   in 
  tlie 
  University 
  of 
  Cambridge 
  "f. 
  

  

  IN 
  an 
  earlier 
  paper 
  J 
  the 
  Author 
  has 
  indicated 
  a 
  relation 
  

   between 
  the 
  integral 
  tabulated 
  by 
  Airy 
  in 
  the 
  course 
  of 
  

   an 
  investigation 
  of 
  the 
  intensity 
  of 
  light 
  near 
  a 
  caustic, 
  and 
  

   the 
  Bessel 
  functions 
  of 
  high 
  order. 
  This 
  relation 
  is 
  approxi- 
  

   mate, 
  and 
  is 
  valid 
  only 
  when 
  the 
  argument 
  of 
  the 
  functions 
  

   is 
  large. 
  Its 
  main 
  utility 
  depends 
  upon 
  the 
  fact 
  that 
  Stokes 
  

   has 
  given 
  extensive 
  tables 
  tor 
  the 
  integral, 
  which 
  may 
  be 
  

   quickly 
  transformed 
  into 
  tables 
  of 
  the 
  Bessel 
  functions 
  of 
  

   high 
  order. 
  But 
  there 
  exists 
  also 
  an 
  exact 
  mathematical 
  

   connexion, 
  which 
  will 
  now 
  be 
  developed. 
  The 
  integral 
  may 
  

   be 
  expressed 
  in 
  terms 
  of 
  the 
  Bessel 
  functions 
  of 
  order 
  ^, 
  

   and 
  certain 
  associated 
  integrals 
  possess 
  the 
  same 
  property. 
  

  

  * 
  Proc. 
  Roy. 
  Soc. 
  vol. 
  xlii. 
  p. 
  80 
  (1887). 
  

   t 
  Communicated 
  by 
  the 
  Author, 
  

   X 
  Phil. 
  Mag. 
  Aug. 
  1908. 
  

  

  