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  XXXIII. 
  A 
  Physical 
  Interpretation 
  of 
  the 
  Bessel 
  Function 
  of 
  

   Zero 
  Order. 
  By 
  J. 
  Hollingworth, 
  21. 
  A., 
  B.Sc.(Eng.) 
  9 
  

   A.C.G.L* 
  

  

  T 
  

  

  HE 
  Bessel 
  function 
  of 
  zero 
  order, 
  regarded 
  as 
  the 
  

   solution 
  of 
  the 
  differential 
  equation 
  

  

  d 
  2 
  y 
  , 
  ldu 
  ~ 
  T 
  

  

  is 
  one 
  which 
  frequently 
  occurs 
  in 
  certain 
  branches 
  of 
  mathe- 
  

   matical 
  physics, 
  and 
  is 
  of 
  great 
  importance. 
  

  

  Now 
  such 
  a 
  solution 
  can 
  be 
  regarded 
  in 
  two 
  different 
  

   ways, 
  one 
  of 
  which 
  has 
  the 
  claim 
  of 
  brevity 
  and 
  neatness, 
  

   the 
  other 
  of 
  more 
  simple 
  physical 
  conception. 
  

  

  This 
  is 
  a 
  wide 
  and 
  far-reaching 
  difference, 
  and 
  depends 
  

   on 
  the 
  point 
  of 
  view 
  of 
  the 
  physicist 
  towards 
  mathematics. 
  

  

  The 
  first 
  point 
  of 
  view 
  is 
  as 
  follows 
  :— 
  * 
  The 
  fundamental 
  

   equations 
  derived 
  from 
  physical 
  principles 
  are 
  found 
  to 
  give 
  

   rise 
  to 
  a 
  differential 
  equation. 
  This 
  equation 
  is 
  then 
  solved 
  

   by 
  purely 
  formal 
  mathematical 
  methods, 
  and 
  the 
  resulting 
  

   solution 
  re-interpreted 
  into 
  physical 
  ideas 
  and 
  applied 
  to 
  the 
  

   problem 
  in 
  question. 
  

  

  This 
  is, 
  no 
  doubt, 
  the 
  simplest 
  and 
  shortest 
  method, 
  but 
  

   it 
  gives 
  the 
  impression 
  that 
  the 
  physics 
  are 
  temporarily 
  lost 
  

   in 
  the 
  mathematics, 
  and 
  the 
  physical 
  process 
  " 
  taking 
  place 
  

   in 
  the 
  solution 
  of 
  the 
  equation," 
  so 
  to 
  speak, 
  is 
  lost. 
  

  

  On 
  the 
  other 
  hand 
  we 
  may 
  proceed 
  as 
  follows. 
  We 
  may 
  

   regard 
  a 
  complex 
  problem 
  as 
  a 
  generalization 
  from 
  the 
  

   simple 
  principles 
  involved. 
  These 
  simple 
  principles 
  can 
  be 
  

   simply 
  expressed 
  mathematically, 
  and 
  consequently, 
  by 
  

   generalizing 
  from 
  these 
  we 
  should 
  be 
  able 
  to 
  build 
  up 
  the 
  

   same 
  solution 
  to 
  the 
  problem 
  as 
  the 
  one 
  obtained 
  by 
  the 
  first 
  

   method, 
  without 
  using 
  any 
  process 
  so 
  apparently 
  arbitrary 
  

   as 
  to 
  write 
  down 
  a 
  solution 
  obtained 
  by 
  non-physical 
  methods 
  

   and 
  say 
  it 
  is 
  what 
  we 
  require. 
  The 
  two 
  methods 
  of 
  course 
  

   amount 
  to 
  the 
  same 
  thing, 
  but 
  to 
  fully 
  appreciate 
  the 
  first 
  

   requires 
  a 
  good 
  grasp 
  of 
  the 
  logical 
  connexion 
  between 
  

   mathematics 
  and 
  physics, 
  and 
  though 
  of 
  greater 
  analytical 
  

   power, 
  is 
  not 
  of 
  the 
  same 
  value 
  for 
  inductive 
  reasoning 
  on 
  

   which 
  at 
  present 
  physical 
  work 
  must 
  largely 
  depend. 
  

  

  In 
  the 
  present 
  paper 
  an 
  attempt 
  will 
  be 
  made 
  to 
  deduce 
  

   the 
  solution 
  of 
  a 
  particular 
  form 
  of 
  the 
  equation 
  I. 
  by 
  both 
  

   methods 
  so 
  as 
  to 
  show 
  their 
  relative 
  values. 
  

  

  The 
  equation 
  which 
  will 
  be 
  considered 
  is 
  the 
  special 
  

  

  * 
  Communicated 
  by 
  the 
  Author. 
  

  

  