﻿Two- 
  Dimensional 
  Motion 
  of 
  Infinite 
  Liquid. 
  127 
  

  

  IfV;>77>0 
  

  

  cot^«-o-cot^(r-r") 
  

  

  = 
  «[*+2«q»(=^ 
  =i 
  ')opA^{i,-t(e-rJ}]i 
  (22) 
  

   and 
  if 
  ?]>"r}' 
  >0 
  

  

  cot^(r-r')-oot^(r-?") 
  

  

  = 
  _4i2smh(^')exp^{i(f-f)-,}, 
  . 
  (23) 
  

  

  where 
  s-1, 
  2, 
  3, 
  ... 
  . 
  The 
  formulae 
  can 
  be 
  verified 
  by- 
  

   noticing 
  that 
  each 
  side 
  of 
  each 
  equality 
  is 
  equivalent 
  to 
  

  

  2i 
  

  

  l-exp^-V-i'Cf-P)} 
  l-exp^{i,+V-i(f-P)} 
  

  

  For 
  77' 
  great 
  the 
  most 
  important 
  terms 
  of 
  {^(f')} 
  2 
  are 
  

   {A(fOF 
  = 
  «o 
  2 
  exp/^J 
  + 
  2A: 
  «:2COS^-^+72-7oj 
  

  

  + 
  2 
  V3 
  exp(-^-^cos^^ 
  / 
  +73-7o)+--- 
  5 
  (24) 
  

  

  the 
  terms 
  decreasing 
  by 
  successive 
  negative 
  powers 
  of 
  

   exp 
  (27T7/A) 
  ; 
  it 
  is 
  to 
  be 
  noted 
  that 
  the 
  functions 
  of 
  | 
  ; 
  

   which 
  multiply 
  these 
  exponentials 
  are 
  sums 
  which 
  may 
  

   contain 
  constant 
  as 
  well 
  as 
  harmonic 
  terms. 
  

  

  In 
  studying 
  the 
  form 
  of 
  the 
  subject 
  of 
  integration 
  in 
  

   formula 
  (21), 
  for 
  great 
  values 
  of 
  rj' 
  y 
  with 
  a 
  view 
  to 
  

   examining 
  the 
  divergence 
  for 
  t->co 
  , 
  the 
  product 
  of 
  the 
  

   series 
  (22) 
  and 
  (24) 
  may 
  be 
  used. 
  For 
  this 
  purpose 
  a 
  term 
  

   of 
  the 
  product 
  may 
  be 
  ignored 
  if 
  its 
  integral 
  with 
  respect 
  

   to 
  (•' 
  through 
  a 
  range 
  X 
  is 
  zero, 
  or 
  if 
  it 
  contains 
  as 
  factor 
  the 
  

   exponential 
  of 
  a 
  negative 
  multiple 
  of 
  r{ 
  j\. 
  By 
  one 
  or 
  other 
  

   of 
  these 
  tests 
  every 
  term 
  is 
  negligible 
  except 
  one, 
  namely, 
  

  

  J 
  

  

  2e# 
  2 
  exp 
  (47r?//\), 
  

  

  ) 
  the 
  integral, 
  at 
  i 
  

   >ect 
  to 
  f), 
  

  

  [i/c 
  2 
  \ 
  2 
  /27r) 
  exp 
  (4tt«/\). 
  

  

  which 
  contributes 
  to 
  the 
  integral, 
  at 
  its 
  upper 
  limit 
  (after 
  

   integration 
  with 
  respect 
  to 
  f), 
  

  

  