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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