But the heights in both sections being equal, the sections are directly as the breadths; Consequently the velocities are reciprocally as the breadths. VI. _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._ 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 the 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. _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 velocities in the narrow passages and open chanel._ 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 be diminished by the height belonging to the velocity, with which the stream arrives at the fall. PROBLEM. _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._ 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. _Principle III._ And 21 ⁄ 25 _c_: _b_∷ _v_: 25_b_ ⁄ 21_c_ _v_ the veloc. _per_ sec. in the water-way between the obstacles. _Princip. V._ Also (2_a_)²: _vv_∷ _a_: _vv_ ⁄ 4_a_ the height fallen to acquire the vel. v. I. & II. And (2_a_)²: (25_b_ ⁄ 21_c_)² × _vv_∷ _a_: (25_b_ ⁄ 21_c_)² × _vv_ ⁄ 4_a_ the height fallen to acquire the vel. 25_b_ ⁄ 21_c_ _v_. I. & II. Then (25_b_ ⁄ 21_c_)² x (_vv_ ⁄ 4_a_) - (_vv_ ⁄ 4_a_) is the measure of the fall required. VII. Or ((25_b_ ⁄ 21_c_)² - 1) × _vv_ ⁄ 4_a_ is a rule, by which the fall may be readily computed.