Read Jan. 19, 1758.
SOME time before the year 1740, the problem about the fall of water, occasioned by the piers of bridges built across a river, was much talked of at London, on account of the fall that it was supposed would be at the new bridge to be built at Westminster. In Mr. Hawksmore’s and Mr. Labelye’s pamphlets, the former published in 1736, and the latter in 1739, the result of Mr. Labelye’s computations was given: but neither the investigation of the problem, nor any rules, were at that time exhibited to the public.
In the year 1742 was published Gardiner’s edition of Vlacq’s Tables; in which, among the examples there prefixed to shew some of the uses of those tables drawn up by the late William Jones, Esq; there are two examples, one shewing how to compute the fall of water at London-bridge, and the other applied to Westminster-bridge: but that excellent mathematician’s investigation of the rule, by which those examples were wrought, was not printed, altho’ he communicated to several of his friends copies thereof. Since that time, it seems as if the problem had in general been forgot, as it has not made its appearance, to my knowlege, in any of the subsequent publications. As it is a problem somewhat curious, tho’ not difficult, and its solution not generally known (having seen four different solutions, one of them very imperfect, extracted from the private books of an office in one of the departments of engineering in a neighbouring nation), I thought it might give some entertainment to the curious in these matters, if the whole process were published. In the following investigation, much the same with Mr. Jones’s, as the demonstrations of the principles therein used appeared to be wanting, they are here attempted to be supplied.
Principles.
I. A heavy body, that in the first second of time has fallen the height of a feet, has acquired such a velocity, that, moving uniformly therewith, will in the next second of time move the length of 2 a feet.
II. The spaces run thro’ by falling bodies are proportional to one another as the squares of their last or acquired velocities.
These two principles are demonstrated by the writers on mechanics.
III. Water forced out of a larger chanel thro’ one or more smaller passages, will have the streams thro’ those passages contracted in the ratio of 25 to 21.
This is shewn in the 36th prop. of the 2d book of Newton’s Principia.
IV. In any stream of water, the velocity is such, as would be acquired by the fall of a body from a height above the surface of that stream.