Philosophical transactions, Vol. L. Part II. For the year 1758. / Giving some account of the present undertakings, studies, and labours, of the ingenious, in many considerable parts of the world.

by Various; Royal Society

This is evident from the nature of motion.

V. The velocities of water thro’ 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; otherwise the bounds would be overflowed.

At that time, the same quantity, which in any time flows thro’ a section in the open chanel, is delivered in equal time thro’ the narrower passages; or the momentum in the narrow passages must be equal to the momentum in the open chanel; or the rectangle under the section of the narrow passages, by their mean velocity, must be equal to the rectangle under the section of the open chanel by its mean velocity.

Therefore the velocity in the open chanel is to the velocity in the narrower passages, as the section of those passages is to the section of the open chanel.

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: