﻿104 
  Prof. 
  Sudhansukumar 
  Banerji 
  on 
  Aerial 
  

  

  Summing 
  the 
  series, 
  we 
  easily 
  find 
  that 
  the 
  vibration 
  on 
  the 
  

   surface 
  of 
  the 
  enveloping 
  sphere 
  

  

  ikct 
  

  

  -.3 
  inches 
  

  

  a 
  Ldr 
  ^r 
  J 
  r=l 
  

  

  = 
  - 
  ~ 
  U 
  a 
  [-496 
  P 
  : 
  (cos 
  0) 
  + 
  3*180 
  P 
  2 
  (cos 
  0) 
  

  

  - 
  1-708 
  P 
  3 
  (cos 
  0) 
  + 
  2-600 
  P 
  4 
  (cos 
  0) 
  + 
  ... 
  ]/** 
  (11) 
  

  

  We 
  have 
  seen 
  that 
  when 
  the 
  vibration 
  on 
  the 
  surface 
  of 
  

   the 
  enveloping 
  sphere 
  is 
  

  

  2 
  ^P, 
  (cos 
  *).«*", 
  

  

  the 
  velocity 
  potential 
  of 
  the 
  wave-disturbance 
  is 
  

  

  Now 
  when 
  r 
  is 
  large, 
  f 
  n 
  (ikr) 
  = 
  1, 
  so 
  that 
  the 
  factor 
  on 
  which 
  

   the 
  relative 
  intensities 
  in 
  various 
  directions 
  depend 
  is 
  

  

  2A 
  P" 
  ( 
  cos 
  0) 
  

   n 
  ~F 
  n 
  (ik.a 
  + 
  b) 
  

  

  Thus 
  if 
  we 
  put 
  this 
  quantity 
  = 
  F 
  + 
  iGr, 
  the 
  intensity 
  of 
  the 
  

   vibrations 
  in 
  various 
  directions 
  is 
  measured 
  by 
  F 
  2 
  + 
  G 
  2 
  . 
  

  

  The 
  distribution 
  of 
  intensities 
  in 
  different 
  directions 
  round 
  

   the 
  spheres 
  will 
  be 
  influenced 
  to 
  a 
  considerable 
  extent 
  by 
  

   the 
  value 
  of 
  the 
  wave-length 
  chosen. 
  If 
  we 
  take 
  k(a 
  + 
  b) 
  = 
  2, 
  

   the 
  wave-length 
  is 
  3-7T 
  inches, 
  and 
  if 
  we 
  take 
  Jc(a 
  + 
  b) 
  = 
  3, 
  

   the 
  wave-length 
  is 
  2-7T 
  inches. 
  From 
  the 
  expression 
  (1) 
  for 
  

   the 
  wave-motion 
  produced 
  by 
  a 
  single 
  sphere 
  undergoing 
  an 
  

   instantaneous 
  change 
  of 
  velocity, 
  it 
  is 
  seen 
  that 
  the 
  wave- 
  

   length 
  to 
  be 
  chosen 
  is 
  of 
  the 
  same 
  order 
  as 
  the 
  circumference 
  

   of 
  the 
  sphere. 
  From 
  this, 
  it 
  appears 
  that 
  for 
  a 
  system 
  of 
  

   two 
  spheres 
  whose 
  radii 
  are 
  1 
  inch 
  and 
  2 
  inches 
  respectively, 
  

   the 
  wave-length 
  to 
  be 
  chosen 
  sbould 
  be 
  some 
  value 
  inter- 
  

   mediate 
  between 
  2tt 
  and 
  47r, 
  probably 
  nearer 
  2it 
  than 
  4-7T 
  ; 
  

   for, 
  in 
  the 
  actual 
  case 
  of 
  impact, 
  the 
  smaller 
  ball 
  which 
  

   would 
  undergo 
  by 
  far 
  the 
  greater 
  change 
  in 
  velocity 
  would 
  

   probably 
  influence 
  the 
  character 
  of 
  the 
  motion 
  to 
  a 
  greater 
  

   extent 
  than 
  the 
  larger 
  sphere. 
  At 
  the 
  same 
  time, 
  it 
  must 
  

   not 
  be 
  forgotten 
  that 
  the 
  analogy 
  between 
  the 
  cases 
  of 
  

   impact 
  and 
  of 
  periodic 
  motion 
  cannot 
  be 
  pushed 
  very 
  far, 
  

   inasmuch 
  as 
  the 
  fluid 
  motion 
  due 
  to 
  impact 
  is 
  undoubtedly 
  

  

  