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sections are directly as the breadths;      (82                                                                 
Consequently the velocities are reciprocally as the breadths.
VI.   [[underline]] In a running stream, the water above any obstacles put therein will rise to such a height, that by its fall the stream may be discharged as fast as it comes. [[/underline]]
    For the same body of water, which flowed in the open chanel, must pass thro' the passages made by the obstacles:
    And the narrower the passages, the swifter will be the velocity of water:
    But the swifter the velocity of the water, the greater is the height, from whence it has descended:
    Consequently the obstacles, which contract the chanel, cause the water to rise against them.
    But the rise will cease, when the water can run off as fast as it comes:
    And this must happen, when, by the fall between the obstacles, the water will acquire a velocity in a reciprocal proportion to that in the open chanel as the breadth of the open chanel is to the breadth of the narrow passages.
VII. [[underline]] The quantity of the fall caused by an obstacle in a running stream is measured by the difference between the heights fallen from to acquire the velocity in the narrow passages and open chanel. [[/underline]]
     For just above the fall, the velocity of the stream is such, as would be acquired by a body falling from a height higher than the surface of the water:
     And at the fall, the velocity of the stream is such, as would be acquired by the fall of a body from a height more elevated than the top of the falling stream; and consequently the real fall is less than this height. 
     Now as the stream comes to the fall with a velocity belonging to a fall above its surface; 
     Consequently the height belonging to the velocity at the fall must by diminished by the height belonging to the velocity, with which the stream arrives at the fall.
PROBLEM.  [[underline]]  In a chanel of running water, whose breadth is contracted by one or more obstacles; the breadth of the chanel, the mean velocity of the whole stream, and the breadth of the water way between the obstacles being given; to find the quantity of the fall occasioned by those obstacles.[[/underline]] 
      Let b = breadth of the chanel in feet.
          v = mean velocity of the water in feet per sec.
          c = breadth of the water-way between the obstacles.
      Now 25:21::c:21/25 c the water-way contracted... [[underline]] Principle [[/underline]] III
      And 21/25c:b::v:25b/21c v the velocity [[underline]] per[[/underline]] sec. in the waterway between the obstacles -------- Princip. V.
      Also (2a)²:vv::a:vv/4a the height fallen to acquire the vel. v.-----------------I. & II.
           And (2a)²:[[(25b/21c)² X vv::a:(25b/21c)² X vv/4a
the height fallen