104 PHILOSOPHICAL TRANSACTIONS. [aNNO 1758. 



5. The velocities of water through different passages of the same height, are 

 reciprocally proportional to their breadths. — For, at some time the water must 

 be delivered as fast as it comes; otherswise the bounds would be overflowed. 

 At that time, the same quantity which in any time flows through a section in 

 the open channel, is delivered in equal time through the narrower passages ; or 

 the momentum in the narrower passages must be equal to the momentum in 

 the open channel ; or the rectangle under the section of the narrower passages, 

 by their mean velocity, must be equal to the rectangle under the section of the 

 open channel by its mean velocity. Therefore the velocity in the open channel, 

 is to the velocity in the narrower passages, as the section of those passages, is 

 to the section of the open channel. But the heights in both sections being 

 equal, the sections are directly as the breadths ; consequently the velocities are 

 reciprocally as the breadths. 



6. In a running stream, the water above any obstacles put in it, 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 channel, must pass 

 through 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 which it has descended : consequently the 

 obstacles which contract the channel, 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 ve- 

 locity in a reciprocal proportion to that in the open channel, as the breadth of the 

 open channel is to the breadth of the narrow passages. 



7. 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 velo- 

 cities in the narrow passages and open channej. — For just above the fall, the ve- 

 locity 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 : 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 channel of running water, whose breadth is contracted by 

 one or more obstacles ; the breadth of the channel, 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. 



