﻿Waves 
  generated 
  by 
  Impact. 
  101 
  

  

  To 
  obtain 
  the 
  equivalent 
  vibrations 
  on 
  the 
  surface 
  of 
  the 
  

   imaginary 
  enveloping 
  sphere, 
  we 
  shall 
  regard 
  the 
  small 
  

   quantity 
  of 
  fluid 
  enclosed 
  by 
  this 
  sphere 
  as 
  practically 
  in- 
  

   compressible, 
  and 
  use 
  the 
  well-known 
  solution 
  by 
  the 
  method 
  

   of 
  successive 
  images 
  for 
  two 
  spheres 
  in 
  an 
  incompressible 
  

   fluid. 
  

  

  We 
  know 
  that 
  the 
  velocity 
  potential 
  due 
  to 
  such 
  a 
  system 
  

   of 
  two 
  spheres 
  in 
  an 
  incompressible 
  fluid 
  can 
  be 
  expressed 
  

   in 
  the 
  form 
  

  

  Tty 
  + 
  U.f,. 
  ....... 
  (5) 
  

  

  where 
  <£ 
  and 
  <\>' 
  are 
  to 
  be 
  determined 
  by 
  the 
  conditions 
  

  

  V 
  2 
  <£ 
  = 
  0, 
  vy=o, 
  

  

  ~- 
  = 
  — 
  co^Ou 
  and 
  ^-5- 
  =0, 
  when 
  j'i=a, 
  

   (to 
  (to 
  

  

  s~^ 
  = 
  — 
  cos 
  #0, 
  and 
  ^~ 
  = 
  0, 
  when 
  r 
  2 
  = 
  b, 
  

   or 
  2 
  or 
  2 
  

  

  r 
  l5 
  r 
  2 
  being 
  radii 
  vectores 
  drawn 
  from 
  A 
  and 
  B. 
  

  

  When 
  <f> 
  and 
  </>' 
  have 
  been 
  determined 
  so 
  as 
  to 
  satisfy 
  

   these 
  conditions, 
  the 
  equivalent 
  vibrations 
  on 
  the 
  surface 
  of 
  

   the 
  imaginary 
  sphere 
  can 
  be 
  taken 
  to 
  be 
  very 
  approximately 
  

   given 
  by 
  the 
  expression 
  

  

  -\vM 
  +Vi 
  mr 
  ..... 
  (6 
  ) 
  

  

  L 
  Or 
  or 
  J 
  r 
  =a+b 
  

  

  The 
  functions 
  <£ 
  and 
  <£', 
  as 
  is 
  well 
  known, 
  can 
  be 
  deter- 
  

   mined 
  by 
  the 
  method 
  of 
  successive 
  images, 
  and 
  if 
  the 
  

   expressions 
  for 
  the 
  velocity 
  potential 
  due 
  to 
  these 
  images 
  be 
  

   all 
  transferred 
  to 
  the 
  coordinates 
  r, 
  6 
  referred 
  to 
  the 
  centre 
  

   C 
  of 
  the 
  enveloping 
  sphere, 
  we 
  easily 
  obtain 
  

  

  2<£ 
  = 
  a 
  3 
  [~l- 
  — 
  ^tts 
  + 
  t- 
  

   L 
  (a 
  + 
  6) 
  (a 
  

  

  {a 
  + 
  b) 
  3 
  ' 
  (a 
  + 
  26) 
  3 
  (2a 
  + 
  26) 
  3 
  

  

  b 
  3 
  "I 
  Px 
  (cos 
  6) 
  

  

  (2a 
  + 
  36) 
  

  

  3 
  r 
  b\a 
  2 
  -hab-b 
  2 
  ) 
  b 
  3 
  (2b 
  2 
  - 
  a 
  2 
  ) 
  b 
  \2a 
  2 
  + 
  ab 
  - 
  2b 
  2 
  ) 
  

   (a 
  + 
  b)' 
  + 
  {a 
  + 
  2by 
  (2a 
  + 
  2by 
  

  

  ,6 
  3 
  (36 
  2 
  -2a 
  2 
  ) 
  -|P 
  2 
  (cos6>) 
  

  

  (2a 
  +36) 
  

  

  ?-.] 
  

  

  