﻿ACADEMY 
  OF 
  SCIENCES. 
  

  

  361 
  

  

  of 
  the 
  tide, 
  from 
  having' 
  its 
  rate 
  of 
  rise, 
  let 
  us 
  next 
  turn 
  to 
  the 
  subject 
  of 
  filling 
  

   a 
  canal 
  or 
  reservoir 
  from 
  tide 
  water. 
  

  

  Let 
  G 
  A 
  M 
  T 
  be 
  the 
  summit 
  reach 
  of 
  a 
  canal, 
  and 
  AG 
  the 
  lock-gate 
  at 
  the 
  

   entrance, 
  through 
  which 
  the 
  tide 
  water 
  is 
  admitted 
  ; 
  J 
  P 
  being 
  the 
  level 
  of 
  the 
  

   ebb 
  or 
  low 
  tide, 
  and 
  T 
  G 
  that 
  of 
  flood 
  or 
  high 
  tide. 
  

  

  Let 
  us 
  suppose 
  that 
  while 
  the 
  tide 
  was 
  rising 
  the 
  distance 
  AG, 
  it 
  had 
  ad- 
  

   vanced 
  in 
  the 
  canal 
  the 
  distance 
  A 
  B 
  : 
  then 
  the 
  surface 
  of 
  the 
  water 
  in 
  the 
  

   canal 
  at 
  the 
  end 
  of 
  the 
  first 
  tide 
  will 
  be 
  indicated 
  by 
  the 
  line 
  GB. 
  

  

  While 
  the 
  tide 
  is 
  at 
  the 
  level 
  T 
  G, 
  the 
  gate 
  A 
  G 
  is 
  closed, 
  and 
  the 
  water 
  in 
  

   the 
  canal 
  will 
  commence 
  to 
  descend 
  from 
  G 
  and 
  advance 
  beyond 
  B. 
  Let 
  us 
  

   suppose 
  that 
  during 
  a 
  given 
  time 
  it 
  has 
  fallen 
  the 
  distance 
  GH, 
  and 
  advanced 
  

   the 
  distance 
  B 
  : 
  then 
  the 
  line 
  G 
  B 
  will 
  conform 
  to 
  H 
  C. 
  

   Let 
  a; 
  = 
  G 
  H, 
  and 
  1/ 
  = 
  B 
  C. 
  

  

  ^sec, 
  ,_ 
  time 
  of 
  falling 
  x, 
  or 
  = 
  time 
  of 
  advancing 
  distance 
  y. 
  

   V 
  = 
  horizontal 
  velocity 
  of 
  tide 
  at 
  the 
  point 
  B 
  ; 
  this 
  can 
  be 
  ascertained 
  

   by 
  the 
  coeflBeient 
  already 
  found. 
  

  

  Let 
  o 
  = 
  A 
  G 
  = 
  the 
  distance 
  the 
  tide 
  has 
  risen, 
  from 
  the 
  moment 
  it 
  com- 
  

   mences 
  to 
  enter 
  the 
  canal 
  until 
  it 
  reaches 
  flood 
  or 
  high 
  tide. 
  

  

  b 
  = 
  AB 
  ^ 
  horizontal 
  distance 
  advanced 
  in 
  the 
  same 
  time. 
  

   Let 
  G 
  n 
  = 
  dx, 
  and 
  Bm 
  = 
  dy. 
  

  

  Let 
  r 
  = 
  mean 
  velocity 
  with 
  which 
  x 
  is 
  described, 
  and 
  z 
  = 
  mean 
  velocity 
  in 
  

   describing 
  y. 
  

  

  Now, 
  because 
  the 
  velocity 
  of 
  a 
  current 
  is 
  as 
  the 
  square 
  root 
  of 
  its 
  inclination, 
  

   ■dx 
  

  

  

  a 
  

   Thi 
  

  

  

  dx 
  

  

  X 
  

  

  velocity 
  at 
  ra. 
  

  

  dy 
  " 
  " 
  ' 
  " 
  ^ 
  b 
  + 
  dij 
  

   may 
  be 
  considered 
  the 
  velocity 
  with 
  which 
  (/// 
  is 
  described, 
  and 
  multi- 
  

  

  plying 
  by 
  dt 
  we 
  get 
  dt 
  v 
  xf 
  ^l~J^\^t 
  ^ 
  dii 
  ^ 
  

   * 
  b 
  +dy^a 
  -^ 
  

  

  .'. 
  ^' 
  \^I~vX---and/rV^«^ 
  

   ^ 
  b 
  + 
  du 
  a 
  ^ 
  b 
  + 
  

  

  dx 
  b 
  

   dy^'n 
  

  

  