﻿108 
  Prof. 
  Sudhansukumar 
  Banerji 
  on 
  Aerial 
  

  

  Summing 
  the 
  series, 
  we 
  find 
  that 
  the 
  vibrations 
  on 
  the 
  

   surface 
  o£ 
  the 
  enveloping 
  sphere, 
  namely 
  

  

  L 
  Or 
  drj 
  r=2 
  inches 
  

  

  can 
  be 
  expressed 
  in 
  the 
  form 
  

  

  i[(U 
  a 
  -U 
  6 
  ) 
  X 
  -2254 
  P^cos 
  6) 
  + 
  (U 
  B 
  + 
  U,) 
  x 
  '3550 
  P 
  2 
  (eos 
  0) 
  

  

  + 
  ( 
  U 
  a- 
  U 
  6) 
  x 
  * 
  3645 
  p 
  s(cos 
  6) 
  + 
  (U 
  a 
  + 
  U 
  6 
  ) 
  x 
  -3080 
  P 
  4 
  (cos 
  0) 
  

  

  + 
  (U 
  fl 
  -U 
  6 
  )x-2325P 
  6 
  (cos^) 
  + 
  &c>^ 
  (16) 
  

  

  If 
  the 
  ball 
  of 
  radius 
  b 
  is 
  four 
  times 
  heavier 
  than 
  the 
  one 
  

   of 
  radius 
  a, 
  we 
  have 
  

  

  a 
  b 
  

  

  So 
  that 
  the 
  vibration 
  on 
  the 
  surface 
  of 
  the 
  enveloping 
  sphere 
  

   is 
  proportional 
  to 
  the 
  expression 
  

  

  •6762 
  P^cos 
  0) 
  + 
  1-7750 
  P 
  2 
  (cos 
  0) 
  + 
  1-0935 
  P 
  3 
  (cos 
  0) 
  

  

  + 
  1-5400 
  P 
  4 
  (cos 
  6) 
  + 
  -6975 
  P 
  5 
  (cos 
  0) 
  + 
  &c. 
  

  

  Now 
  taking 
  k(a 
  + 
  b)=l 
  9 
  which 
  will 
  give 
  a 
  wave-length 
  

   equal 
  to 
  the 
  circumference 
  of 
  the 
  enveloping 
  sphere, 
  we 
  get 
  

   (neglecting 
  a 
  constant 
  factor) 
  

  

  F= 
  -13524 
  P,(cos 
  0)- 
  -04987 
  P 
  2 
  (cos 
  0)— 
  0074 
  P 
  3 
  (cos 
  0) 
  

   + 
  •0007 
  P 
  4 
  (cos 
  0)+ 
  Ac. 
  

  

  G= 
  - 
  '0676 
  Px(cos 
  0) 
  - 
  -0798 
  P 
  2 
  (cos 
  0) 
  + 
  -0047 
  P 
  3 
  (cos 
  0) 
  

   + 
  -0012P 
  4 
  (cos6>)- 
  &c. 
  . 
  . 
  (17) 
  

  

  The 
  values 
  of 
  F 
  and 
  G, 
  and 
  of 
  F 
  2 
  + 
  G 
  2 
  in 
  different 
  directions 
  

   obtained 
  from 
  the 
  preceding 
  expressions 
  are 
  shown 
  in 
  

   Table 
  III. 
  

  

  The 
  values 
  of 
  (F 
  2 
  +G 
  2 
  ) 
  shown 
  in 
  Table 
  III. 
  have 
  been 
  

   plotted 
  in 
  polar 
  coordinates 
  and 
  are 
  shown 
  in 
  fig. 
  6. 
  It 
  is 
  

   seen 
  that 
  the 
  maximum 
  intensity 
  in 
  the 
  direction 
  of 
  the 
  

   heavier 
  ball 
  is 
  greater 
  than 
  that 
  in 
  the 
  direction 
  of 
  the 
  

   lighter 
  one. 
  

  

  The 
  experimental 
  curve 
  of 
  intensity 
  of 
  sound 
  due 
  to 
  

   impact 
  of 
  a 
  sphere 
  of 
  wood, 
  diameter 
  2\ 
  inches, 
  with 
  a 
  billiard 
  

   ball 
  of 
  nearly 
  the 
  same 
  size 
  is 
  shown 
  in 
  fig. 
  7. 
  It 
  is 
  found 
  

  

  