﻿Problems 
  connected 
  with 
  the 
  Circular 
  Arc. 
  401 
  

  

  neater 
  form. 
  The 
  elimination 
  of 
  f 
  between 
  (3) 
  and 
  (12) 
  is 
  

   rather 
  lengthy, 
  but 
  not 
  difficult. 
  AVe 
  obtain 
  

  

  tzic' 
  + 
  Lwiz+iXze-'P-Le^+iz 
  + 
  i) 
  2 
  - 
  sin 
  2 
  |(*r*+«?'*) 
  2 
  =0, 
  (13) 
  

   from 
  which 
  we 
  have 
  as 
  the 
  value 
  of 
  iv 
  

  

  (z+i)(ze-*-ie#) 
  + 
  {ze-* 
  + 
  ie'f 
  i 
  ) 
  s 
  /(z 
  2 
  + 
  2i,z 
  cos 
  gl-1) 
  M 
  _. 
  

  

  where 
  the 
  same 
  branch 
  of 
  the 
  multiform 
  function 
  is 
  to 
  be 
  

   taken 
  as 
  in 
  § 
  3 
  above. 
  The 
  real 
  and 
  imaginary 
  parts 
  of 
  w 
  

   give 
  the 
  velocity 
  potential 
  and 
  stream 
  function 
  of 
  the 
  motion 
  

   due 
  to 
  the 
  disturbance 
  of 
  a 
  uniform 
  stream 
  by 
  the 
  lamina, 
  

   the 
  undisturbed 
  velocity 
  of 
  the 
  stream 
  being 
  unity 
  and 
  its 
  

   direction 
  inclined 
  at 
  an 
  angle 
  /3 
  to 
  the 
  axis 
  of 
  x. 
  The 
  

   corresponding 
  values 
  for 
  the 
  motion 
  of 
  the 
  lamina 
  in 
  a 
  

   liquid 
  at 
  rest 
  at 
  infinity 
  are 
  obtained 
  by 
  adding 
  ze~^ 
  to 
  the 
  

   above 
  value 
  of 
  to. 
  Owing 
  to 
  the 
  computation 
  of 
  square 
  roots 
  

   of 
  complex 
  quantities 
  being 
  a 
  laborious 
  process 
  in 
  practice, 
  

   the 
  stream-lines 
  have 
  not 
  been 
  plotted, 
  though 
  of 
  course. 
  

   (13) 
  could 
  be 
  used 
  if 
  it 
  were 
  required 
  to 
  do 
  so. 
  In 
  the 
  

   above 
  the 
  radius 
  of 
  the 
  arc. 
  and 
  the 
  velocity, 
  have 
  been 
  taken 
  

   as 
  unity. 
  The 
  only 
  modifications 
  to 
  include 
  the 
  case 
  of 
  a 
  

   velocity 
  U 
  and 
  radius 
  a 
  are 
  that 
  in 
  (14) 
  a 
  factor 
  U 
  must 
  be 
  

   introduced 
  on 
  the 
  left-hand 
  side, 
  and 
  z, 
  w 
  replaced, 
  by 
  z/a, 
  

   iv/a 
  respectively. 
  

  

  § 
  6. 
  To 
  obtain 
  the 
  "impulse 
  " 
  of 
  the 
  second 
  of 
  the 
  above 
  

   motions, 
  we 
  have 
  only 
  to 
  expand 
  in 
  terms 
  of 
  1/z 
  and 
  take 
  

   — 
  2irp 
  times 
  the 
  coefficient 
  of 
  1/z 
  (/>= 
  density 
  of 
  liquid). 
  

   The 
  required 
  coefficient 
  is 
  found 
  to 
  be 
  

  

  — 
  %{e 
  l 
  P(l— 
  cos 
  «) 
  — 
  ^sin 
  2 
  ae~ 
  tli 
  }, 
  

  

  so 
  for 
  the 
  components 
  of 
  impulse 
  we 
  have 
  

  

  (15) 
  

  

  X 
  = 
  2irp 
  cos 
  /3 
  -I 
  sin 
  2 
  * 
  — 
  -, 
  sin 
  2 
  at 
  = 
  2irp 
  cos 
  /3 
  sin 
  1 
  - 
  ; 
  \ 
  

  

  Y 
  =27rp 
  sin 
  fi 
  < 
  sin 
  2 
  ~ 
  + 
  .sin 
  2 
  a 
  > 
  = 
  27r/jsinj8siii 
  2 
  ^l 
  1 
  + 
  cos 
  2 
  ^)- 
  ) 
  

  

  On 
  introducing 
  the 
  velocity 
  U 
  and 
  radius 
  a, 
  these 
  become 
  

  

  brpa 
  2 
  sin*£.U 
  cos 
  & 
  X 
  = 
  27r/™ 
  2 
  sin 
  2 
  *('l.+ 
  cos 
  2 
  -*) 
  . 
  U 
  sin 
  0. 
  (1;V 
  

  

  AVe 
  may 
  proceed 
  to 
  the 
  limiting 
  case 
  of 
  a 
  plane 
  lamina 
  of 
  

   width 
  2b 
  by 
  making 
  a->x> 
  and 
  a->0 
  in 
  such 
  a 
  manner 
  that 
  

   a 
  sin 
  «••>/>. 
  We 
  obtain 
  

  

  X 
  = 
  0, 
  Y 
  = 
  r/>6 
  2 
  Usin/3, 
  .... 
  (16) 
  

  

  