﻿362 
  

  

  PROCEEDINGS 
  OF 
  THE 
  CALIFORNIA 
  

  

  abt'^ 
  v2 
  — 
  bfi 
  v'i 
  rdx 
  = 
  ahij- 
  + 
  aifi 
  dy. 
  Substitute 
  for 
  dx, 
  and 
  there 
  resu'ts 
  

  

  abti 
  «'- 
  — 
  bt- 
  v^- 
  rdt 
  = 
  abf- 
  + 
  ai/2 
  dy. 
  Integrate, 
  and 
  we 
  get 
  

  

  ahfi 
  v'i 
  — 
  b 
  V' 
  r 
  1^ 
  = 
  aby'i 
  + 
  aif 
  := 
  abl!^ 
  z^ 
  + 
  at'^ 
  z"*- 
  y 
  

  

  abv^ 
  — 
  bv'^ 
  X 
  = 
  a6c2 
  

   3 
  

  

  + 
  ac-' 
  

  

  Hence, 
  iv 
  

  

  \fb( 
  a 
  — 
  K 
  ^ 
  

   ^ 
  a\ 
  b 
  + 
  3y 
  

  

  ,No 
  ic. 
  

  

  Again 
  : 
  let 
  MJEG 
  represent 
  a 
  cross 
  section 
  

   of 
  the 
  canal. 
  

  

  w 
  = 
  width 
  at 
  bottom. 
  

   a 
  = 
  depth 
  as 
  before. 
  

   n 
  == 
  ratio 
  of 
  slopes 
  of 
  banks. 
  

   Then 
  the 
  quantity 
  of 
  water 
  received 
  by 
  the 
  

   summit 
  reach 
  from 
  one 
  tide, 
  whose 
  longitud- 
  

   inal 
  section 
  is 
  AGB 
  and 
  cross 
  section 
  MJEG, 
  

   will 
  be 
  

  

  U+-3-; 
  

  

  X 
  ab 
  

  

  When 
  G 
  has 
  descended 
  to 
  H 
  

   and 
  hence 
  

  

  then 
  a 
  becomes 
  a 
  — 
  x, 
  and 
  b 
  becomes 
  b 
  -{- 
  y, 
  

  

  (f 
  +-T-) 
  X 
  .6 
  = 
  (|+fc|>J) 
  X 
  („ 
  _ 
  .) 
  X 
  (i 
  + 
  ,) 
  . 
  . 
  . 
  NO 
  2=. 
  

  

  From 
  these 
  two 
  equations, 
  when 
  any 
  one 
  of 
  the 
  quantities 
  x, 
  y, 
  L 
  is 
  given 
  , 
  

   the 
  other 
  two 
  can 
  be 
  found. 
  

  

  Considering 
  the 
  movement 
  of 
  the 
  tide 
  within 
  the 
  summit 
  reach 
  of 
  the 
  canal, 
  

   it 
  is 
  manifest 
  that 
  when 
  the 
  reach 
  is 
  long 
  as 
  in 
  the 
  instance 
  I 
  have 
  referred 
  to, 
  

   the 
  quantity 
  of 
  water 
  poured 
  into 
  it 
  by 
  one 
  tide 
  cannot 
  have 
  come 
  to 
  a 
  level 
  

   before 
  the 
  next 
  succeeding 
  tide 
  commences 
  to 
  enter 
  the 
  canal. 
  Let 
  us 
  suppose 
  

   that 
  H 
  (fig. 
  2) 
  is 
  the 
  point 
  at 
  which 
  this 
  succeeding 
  tide 
  begins 
  to 
  enter 
  : 
  then 
  

   the 
  time 
  occupied 
  by 
  the 
  tide 
  in 
  flowing 
  into 
  the 
  canal 
  is 
  limited 
  to 
  the 
  time 
  the 
  

   tide 
  takes 
  to 
  rise 
  from 
  H 
  to 
  G. 
  

  

  Let 
  g 
  = 
  entire 
  rise 
  of 
  tide, 
  = 
  JT, 
  or 
  difference 
  of 
  level 
  between 
  high 
  and 
  

   low 
  tide. 
  

  

  c 
  = 
  mean 
  vertical 
  velocity 
  of 
  tide. 
  

  

  Then 
  

  

  = 
  the 
  time 
  that 
  has 
  

  

  from 
  previous 
  high 
  tide 
  until 
  

  

  water 
  commences 
  again 
  to 
  enter 
  the 
  canal 
  

   Equation 
  N° 
  1° 
  now 
  becomes 
  

   2g- 
  

  

  ^sec. 
  

  

  X 
  

  

  y\fb(a--yix\ 
  _ 
  

   ^ 
  a\b 
  + 
  y,yj 
  

  

  y 
  N^P 
  

  

  we 
  can 
  obtain 
  x 
  and 
  y, 
  and 
  we 
  are 
  thus 
  

  

  From 
  Equations 
  N^ 
  l"" 
  and 
  N° 
  2 
  

   enabled 
  to 
  ascertain 
  the 
  elevation 
  at 
  which 
  every 
  tide 
  commences 
  to 
  enter 
  the 
  

  

  