TRACT 38.

PROBLEMS.

371

PROBLEM XXXI.

To determine the Fall of the Water under the Arches of a

Bridge.

The effects of obstacles placed in a current of water, suchas the piers of a bridge, are, a sudden steep descent, and anincrease of velocity in the stream of water, just under thearches, more or less in proportion to the quantity 7 of the ob-struction and velocity of the current: being very small andhardly perceptible where the arches are large and the piersfew or small, but in a high and extraordinary degree atLondon-bridge, and some others, where the piers and thesterlings are so very large, in proportion to the arches. Thisis the case, not only in such streams as run always the sameway, but in tide rivers also, both upward and downward, butmuch less in the former than in the latter. During the timeof flood, when the tide is flowing upward, the rise of thewater is against the under side of the piers ; but the differ-ence between the two sides gradually diminishes as the tideflows less rapidly towards the concluson of the flood. Whenthis has attained its full height, and there is no longer any'current, but a stillness prevails in the water for a short time,iflc surface assumes an equal level, both above and belowbridge. But, as soon as the tide begins to ebb or returnagain, the resistance of the piers against the stream, and thecontraction of the waterway, cause a rise of the surface aboveand under the arches, with a fall and a more rapid descentin the contracted stream just below. The quantity of thisrise, and of the consequent velocity below, keep both gra-dually increasing, as the tide continues ebbing, till at quitelow water, when the stream or natural current being thequickest, the fall under the arches is the greatest. And it isthe quantity of this fall which it is the object of this problemto determine.

Now, the motion of free running water is the consequence

SB b C