﻿500 
  Mr. 
  W. 
  G. 
  Bickley 
  on 
  Two-Dimensional 
  

  

  5. 
  The 
  velocity 
  at 
  any 
  point 
  o£ 
  the 
  curve 
  is 
  found 
  from 
  

   (3) 
  to 
  be 
  given 
  bv 
  

  

  v*=2 
  9 
  F{z). 
  

  

  We 
  see 
  from 
  (2) 
  that 
  z—li 
  makes 
  P(Y) 
  = 
  0. 
  Thus 
  the 
  

   level 
  of 
  no 
  velocity 
  is 
  given 
  by 
  z 
  = 
  h. 
  

  

  6. 
  To 
  study 
  the 
  nature 
  of 
  the 
  surface 
  we 
  observe 
  at 
  the 
  

   outset 
  that, 
  when 
  yLt 
  = 
  0, 
  i. 
  e. 
  when 
  the 
  surface 
  is 
  smooth, 
  

   the 
  limiting 
  form 
  of 
  (2) 
  becomes 
  

  

  P(z) 
  = 
  h-z, 
  

  

  and 
  taking 
  A 
  = 
  0. 
  we 
  obtain* 
  

  

  z{x 
  2 
  + 
  f) 
  = 
  tanci.km-^. 
  

  

  Now 
  consider 
  the 
  general 
  equation 
  (1). 
  Then 
  we 
  find 
  

   that 
  (i.) 
  the 
  section 
  by 
  the 
  cylinder 
  x 
  2 
  -\-y 
  2 
  =l 
  is 
  the 
  helix 
  

   x 
  = 
  cos 
  (z 
  cot 
  a) 
  , 
  y 
  — 
  sin 
  (z 
  cot 
  a) 
  ; 
  (ii.) 
  the 
  section 
  by 
  the 
  hori- 
  

   zontal 
  plane 
  z 
  = 
  h 
  is 
  the 
  straight 
  line 
  y 
  = 
  x 
  tan 
  (h 
  cot 
  a), 
  z 
  = 
  h 
  ; 
  

   (iii.) 
  the 
  other 
  horizontal 
  sections 
  are 
  spirals 
  of 
  the 
  form 
  

   r 
  2 
  = 
  acj) 
  + 
  b 
  ; 
  (iv.) 
  generally, 
  sections 
  by 
  vertical 
  circular 
  

   cylinders 
  are 
  different 
  from 
  helices 
  ; 
  (v.) 
  the 
  surface 
  is 
  not 
  

   a 
  minimal 
  one 
  like 
  the 
  helicoid. 
  Hence 
  we 
  conclude 
  that, 
  

   although 
  for 
  small 
  values 
  of 
  fi 
  the 
  surface 
  has 
  nearly 
  the 
  

   same 
  shape 
  as 
  the 
  surface 
  discussed 
  by 
  Catalan, 
  for 
  fairly 
  

   large 
  values 
  of 
  /x 
  it 
  differs 
  essentially 
  from 
  Catalan's 
  surface 
  

   as 
  well 
  as 
  from 
  the 
  helicoid. 
  

  

  I 
  wish 
  to 
  express 
  my 
  thanks 
  to 
  Prof. 
  Prasad 
  for 
  his 
  kind 
  

   interest 
  in 
  the 
  paper. 
  

  

  LXT. 
  Two-Dimensional 
  Motion 
  of 
  an 
  Infinite 
  Liquid. 
  

  

  By^N. 
  G. 
  Bickley,' 
  B.Scj 
  

  

  § 
  1. 
  TN 
  a 
  recent 
  paper 
  (Phil. 
  Mag. 
  (6) 
  xxxv. 
  no. 
  205, 
  

  

  JL 
  p. 
  119, 
  Jan. 
  1918) 
  Dr. 
  J. 
  G. 
  Leathern 
  has 
  shown 
  

  

  how 
  to 
  determine 
  the 
  motion 
  in 
  two 
  dimensions 
  of 
  an 
  infinite 
  

  

  liquid 
  occupying 
  the 
  space 
  outside 
  a 
  solid 
  body 
  bounded 
  by 
  

  

  a 
  closed 
  curve 
  or 
  polygon, 
  due 
  to 
  prescribed 
  motion 
  of 
  the 
  

  

  boundary. 
  The 
  method 
  used 
  depends 
  on 
  the 
  use 
  of 
  periodic 
  

  

  conform 
  al 
  transformations 
  whereby 
  the 
  doubly 
  connected 
  

  

  space 
  outside 
  the 
  boundary 
  is 
  transformed 
  into 
  a 
  semi-infinite 
  

  

  rectangle. 
  The 
  solution 
  for 
  the 
  case 
  of 
  translatory 
  motion 
  

  

  is 
  neat 
  and 
  immediate, 
  but 
  this 
  can 
  hardly 
  be 
  said 
  of 
  the 
  

  

  solution 
  in 
  the 
  case 
  of 
  rotation, 
  although 
  it 
  is 
  perfectly 
  

  

  * 
  This 
  is 
  the 
  surface 
  discussed 
  by 
  Catalan 
  {Journal 
  de 
  mathematiques, 
  

   ser. 
  1, 
  tome 
  xi.). 
  

   f 
  Communicated 
  by 
  the 
  Author. 
  

  

  