﻿100 
  Prof. 
  Sudhansukumar 
  Banerji 
  on 
  Aerial 
  

  

  different 
  directions, 
  we 
  may 
  consider 
  the 
  analogous 
  acoustical 
  

   problem 
  of 
  two 
  rigid 
  spheres 
  nearly 
  in 
  contact, 
  which 
  execute 
  

   small 
  oscillations 
  to 
  and 
  fro 
  on 
  the 
  line 
  of 
  their 
  centres. 
  

   This 
  problem 
  may 
  be 
  mathematically 
  formulated 
  and 
  approx- 
  

   imately 
  solved 
  in 
  the 
  following 
  manner 
  : 
  — 
  

   Given 
  prescribed 
  vibrations 
  

  

  ikct 
  

  

  U 
  cos 
  X 
  . 
  e 
  ikct 
  and 
  U 
  6 
  cos 
  6 
  2 
  . 
  e 
  

  

  on 
  the 
  surfaces 
  of 
  two 
  spheres 
  of 
  radii 
  a 
  and 
  b 
  nearly 
  in 
  

   contact, 
  it 
  is 
  required 
  to 
  determine 
  the 
  velocity 
  potential 
  of 
  

   the 
  wave-motion 
  started 
  and 
  the 
  distribution 
  of 
  intensities 
  

   round 
  the 
  spheres, 
  where 
  1 
  and 
  6 
  2 
  are 
  the 
  angles 
  measured 
  

   at 
  the 
  centres 
  A 
  and 
  B 
  of 
  the 
  two 
  spheres 
  in 
  opposite 
  senses 
  

   from 
  the 
  line 
  joining 
  the 
  centres. 
  

  

  Supposing, 
  now, 
  that 
  an 
  imaginary. 
  sphere 
  is 
  constructed 
  

   which 
  is 
  of' 
  just 
  sufficient 
  radius 
  to 
  envelop 
  the 
  two 
  actual 
  

   spheres 
  (touching 
  them 
  externally), 
  it 
  is 
  possible 
  from 
  a 
  

   consideration 
  of 
  the 
  nature 
  of 
  the 
  motion 
  that 
  takes 
  place 
  in 
  

   the 
  immediate 
  neighbourhood 
  of 
  the 
  two 
  spheres, 
  to 
  deter- 
  

   mine 
  the 
  aerial 
  vibration 
  on 
  the 
  surface 
  of 
  the 
  imaginary 
  

   sphere 
  which 
  would 
  produce 
  on 
  the 
  external 
  atmosphere 
  the 
  

   same 
  effect 
  as 
  the 
  vibrations 
  on 
  the 
  surfaces 
  of 
  the 
  real 
  

   spheres 
  A 
  and 
  B. 
  When 
  the 
  equivalent 
  vibration 
  on 
  the 
  

   surface 
  of 
  the 
  enveloping 
  sphere 
  has 
  been 
  obtained, 
  we 
  can, 
  

   by 
  the 
  use 
  of 
  the 
  well-known 
  solution 
  for 
  a 
  single 
  sphere, 
  at 
  

   once 
  determine 
  the 
  wave-motion 
  at 
  any 
  external 
  point. 
  

  

  The 
  radius 
  of 
  the 
  enveloping 
  sphere 
  is 
  evidently 
  a 
  + 
  b, 
  

   and 
  its 
  centre 
  is 
  at 
  a 
  point 
  C, 
  such 
  that 
  BO 
  = 
  a 
  and 
  CA 
  = 
  6. 
  

  

  If 
  the 
  point 
  C 
  be 
  taken 
  as 
  origin, 
  and 
  if 
  the 
  equivalent 
  

   vibration 
  on 
  the 
  surface 
  of 
  the 
  enveloping 
  sphere 
  be 
  expressed 
  

   by 
  the 
  series 
  

  

  2AJP 
  n 
  (cos0V*<*, 
  (2) 
  

  

  where 
  A 
  n 
  's 
  are 
  known 
  constants, 
  the 
  velocity 
  potential 
  of 
  

   the 
  wave-motion 
  is 
  given 
  by 
  

  

  (a 
  + 
  b)*- 
  *(«*-•+«+») 
  ^ 
  AJV, 
  (cos 
  0) 
  

  

  where 
  

  

  f 
  (ikr 
  ) 
  - 
  1 
  + 
  "("+ 
  1 
  ) 
  + 
  (*-lM* 
  + 
  l)(tt 
  + 
  2) 
  + 
  

   • 
  /wW_i+ 
  2.ikr 
  + 
  2.1. 
  (ikr) 
  2 
  + 
  "' 
  

  

  1.2.3...2n 
  

   •* 
  ,+ 
  2.4.6...2n.(t£r)"' 
  

  

  VJikr) 
  = 
  (1 
  + 
  ikr)f 
  n 
  (ikr) 
  - 
  ikr/J 
  (ikr) 
  . 
  . 
  . 
  (4) 
  

  

  