﻿736 
  Prof. 
  C. 
  H. 
  Lees 
  on 
  the 
  Resistance 
  of 
  a 
  

  

  amplitude, 
  we 
  obtain 
  

  

  cos-^— 
  , 
  , 
  sin 
  -^— 
  \ 
  

  

  :anh 
  ( 
  + 
  j 
  ) 
  — 
  i 
  arctan 
  I 
  + 
  j— 
  \ 
  + 
  h 
  

  

  V 
  coshlogvy/^ 
  7 
  V 
  sinhlog^/^ 
  

  

  e 
  a 
  -e 
  1 
  . 
  e 
  a 
  -e 
  Y 
  

  

  1 
  r 
  / 
  cos 
  "^— 
  \ 
  . 
  / 
  sm 
  ~ 
  2~~ 
  \ 
  l 
  

  

  — 
  ■{ 
  argtanh 
  + 
  7 
  — 
  ) 
  — 
  ^ 
  arctan 
  ( 
  ± 
  — 
  ) 
  + 
  hnr 
  y 
  

  

  VaX 
  \~ 
  cosh 
  log* 
  /-^ 
  V 
  sinhlog./^/ 
  

  

  where 
  m 
  and 
  ?z 
  are 
  integers. 
  The 
  alternatives 
  tanh 
  and 
  coth 
  

   of 
  equation 
  (2) 
  both 
  lead 
  to 
  the 
  same 
  result. 
  

  

  On 
  determining 
  the 
  values 
  of 
  the 
  constants 
  A 
  and 
  a 
  so 
  as 
  

   to 
  make 
  ?/ 
  = 
  from 
  £= 
  — 
  go 
  to 
  f 
  = 
  — 
  a, 
  y 
  = 
  c-b 
  from 
  f= 
  — 
  1 
  

   to 
  f 
  =0, 
  y 
  — 
  c 
  from 
  f 
  = 
  to 
  £=co 
  , 
  we 
  find 
  that 
  the 
  negative 
  

   sign 
  must 
  be 
  assigned 
  to 
  the 
  term 
  in 
  the 
  small 
  brackets, 
  that 
  

   A=c/7r, 
  V 
  'a 
  = 
  c/6, 
  and 
  the 
  equations 
  become 
  

  

  . 
  2fc 
  /tanhy 
  / 
  f+1 
  2c 
  /tanh\ 
  / 
  ~?+T~ 
  

  

  8 
  7 
  * 
  + 
  * 
  = 
  * 
  ar 
  s 
  Vcoth 
  ; 
  i 
  v 
  ?t?/p 
  " 
  ? 
  arg 
  v 
  coth 
  V 
  ?+^ 
  2 
  

  

  cos 
  — 
  t 
  — 
  sin 
  — 
  ~ 
  — 
  

  

  = 
  — 
  { 
  ar 
  otanh 
  ==== 
  — 
  i 
  arctan 
  

  

  7T- 
  & 
  

  

  cosh 
  log 
  J^/5 
  sinh 
  logf 
  ^g 
  

  

  cos 
  — 
  ~— 
  - 
  sin 
  „ 
  

  

  — 
  — 
  Sargtanh 
  — 
  — 
  — 
  j 
  — 
  —i 
  arctan 
  — 
  j- 
  f* 
  (3) 
  

  

  " 
  [ 
  cosh 
  log 
  \J^ 
  a 
  sinhlogy^ 
  

  

  where 
  the 
  arctan 
  terms 
  are 
  to 
  be 
  taken 
  within 
  the 
  limits 
  

   and 
  7r. 
  

  

  The 
  curves 
  in 
  the 
  z 
  plane 
  into 
  which 
  this 
  equation 
  con- 
  

   verts 
  the 
  orthogonal 
  system 
  of 
  circles 
  O 
  1 
  — 
  a 
  = 
  constant, 
  and 
  

   rJr 
  a 
  = 
  constant, 
  of 
  the 
  f 
  plane, 
  are 
  shown 
  in 
  fig. 
  3 
  for 
  the 
  

   case 
  c/6 
  = 
  2. 
  They 
  may 
  be 
  considered 
  either 
  as 
  the 
  potential 
  

   and 
  stream 
  curves 
  for 
  a 
  source 
  at 
  1 
  and 
  an 
  equal 
  sink 
  at 
  A, 
  

  

  The 
  term 
  argtanh 
  cos 
  ^ 
  w 
  _ 
  ma 
  y 
  ^ 
  e 
  more 
  familiar 
  in 
  the 
  form 
  

  

  cosh 
  log 
  p 
  + 
  cos 
  p 
  2 
  4-2pcos 
  0+1 
  

  

  2 
  lo 
  £ 
  cosh 
  log 
  p- 
  cos 
  = 
  3 
  log 
  p 
  2 
  -2p 
  cos 
  0+1* 
  

  

  