﻿126 
  Dr. 
  J. 
  G. 
  Leathern 
  on 
  the 
  

  

  satisfied. 
  The 
  equations 
  of 
  continuity 
  and 
  of 
  vorticity 
  are 
  

   respectively 
  

  

  W 
  + 
  ^~ 
  0, 
  §|-^ 
  = 
  2o, 
  wm 
  s 
  , 
  • 
  (20) 
  

  

  when 
  f 
  is 
  inside 
  the 
  area 
  of 
  integration. 
  

  

  The 
  testing 
  o£ 
  these 
  equalities 
  involves 
  the 
  differentiation 
  

   of 
  the 
  integrals 
  of 
  formulae 
  (18) 
  and 
  (19), 
  and 
  this 
  cannot 
  

   be 
  done 
  by 
  the 
  ordinary 
  rule 
  of 
  differentiation 
  under 
  the 
  

   sign 
  of 
  integration, 
  since 
  that 
  would 
  yield 
  semi-convergent 
  

   integrals. 
  It 
  is, 
  however, 
  easy 
  to 
  apply 
  the 
  method 
  of 
  

   differentiation 
  explained 
  in 
  the 
  Cambridge 
  Tract 
  on 
  ' 
  Volume 
  

   and 
  Surface 
  Integrals 
  used 
  in 
  Physics/ 
  articles 
  21 
  and 
  23, 
  

   and 
  it 
  is 
  then 
  readily 
  verified 
  that 
  u 
  t 
  and 
  v 
  t 
  satisfy 
  both 
  

   conditions. 
  

  

  A 
  single 
  compact 
  formula 
  giving 
  both 
  u 
  t 
  and 
  v 
  t 
  is 
  

  

  ft) 
  i"i 
  X 
  

  

  vt 
  + 
  iut— 
  z- 
  

  

  £i 
  x 
  {Mr)} 
  2 
  [cot^(?-?o-oot^(r-r')] 
  <*r*»'- 
  (21) 
  

  

  The 
  integral 
  on 
  the 
  right-hand 
  side 
  has 
  the 
  appearance 
  of 
  

   being 
  a 
  function 
  of 
  the 
  complex 
  variable 
  f 
  ; 
  but 
  this 
  appear- 
  

   ance 
  is 
  deceptive, 
  for 
  if 
  v 
  and 
  u 
  were 
  conjugate 
  functions 
  

   there 
  would 
  be 
  no 
  vorticity 
  *. 
  

  

  9. 
  The 
  next 
  step 
  that 
  suggests 
  itself 
  is 
  a 
  passage 
  to 
  limits 
  

   for 
  t 
  infinitely 
  great. 
  This 
  is 
  feasible 
  in 
  the 
  case 
  of 
  v 
  t 
  , 
  but 
  

   the 
  integral 
  representing 
  u 
  t 
  proves 
  to 
  be 
  divergent. 
  

  

  It 
  will 
  be 
  shown 
  that 
  this 
  can 
  be 
  remedied 
  by 
  adding 
  to 
  u 
  t 
  , 
  

   before 
  passage 
  to 
  limit, 
  a 
  suitable 
  function 
  of 
  t 
  which 
  does 
  

   not 
  involve 
  f 
  or 
  77. 
  An 
  addition 
  to 
  u 
  t 
  means 
  simply 
  the 
  

   superposition 
  ol 
  an 
  irrotational 
  motion 
  with 
  circulation 
  

   round 
  the 
  fixed 
  boundary. 
  This 
  will 
  serve 
  to 
  cancel 
  an 
  

   undesired 
  circulation 
  at 
  infinity 
  in 
  the 
  motion 
  defined 
  by 
  

   ut 
  and 
  Vf 
  

  

  No 
  corresponding 
  addition 
  to 
  v 
  t 
  need 
  be 
  or 
  could 
  be 
  

   made. 
  

  

  It 
  being 
  necessary 
  to 
  consider 
  not 
  only 
  the 
  convergence 
  

   of 
  the 
  u 
  and 
  v 
  integrals 
  but 
  also 
  the 
  forms 
  to 
  which 
  these, 
  

   regarded 
  as 
  functions 
  of 
  f 
  and 
  97, 
  tend 
  for 
  97 
  very 
  great 
  and 
  

   positive, 
  it 
  is 
  important 
  to 
  notice 
  two 
  expansions 
  of 
  the 
  

   function 
  which 
  appears 
  in 
  square 
  brackets 
  in 
  formula 
  (21). 
  

  

  * 
  On 
  this 
  point 
  compare 
  the 
  writer's 
  note 
  l< 
  On 
  Functionality 
  of 
  a 
  

   Complex 
  Variable 
  " 
  in 
  the 
  ' 
  Mathematical 
  Gazette/ 
  early 
  in 
  1918. 
  

  

  