VOL. L.] PHILOSOPHICAL TRANSACTIONS. 1 QS 



LXin. Concerning the Fall of Water under Bridges. By Mr. J. Robertson. 



F. R. S. p. 492. 



Some time before the year 1740, the problem about the fall of water, oc- 

 casioned by the piers of bridges built across a river, was much spoken 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 1 742 was published Gardiner's edition of Vlacq's Tables ; in which 

 among the examples there prefixed to show some of the uses of those tables 

 drawn up by the late William Jones, Esq. there are 2 examples, one showing 

 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, though he com- 

 municated copies of it to several of his friends. Since that time, it seems as if 

 the problem had in general been forgotten, as it has not made its appearance 

 in any of the subsequent publications. As it is a problem somewhat curious, 

 though not difficult, and its solution not generally known (having seen 4 dif- 

 ferent 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,) 

 Mr. R. thought it might give some entertainment to the curious in these mat- 

 ters, if the whole process were published. In the following investigation, much 

 the same with Mr. Jones's, as the demonstrations of the principles used appeared 

 to be wanting, they are here attempted to be supplied. 



PRINCIPLES. 



1 . 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 with it, will in the 

 next second of time move the length of 2a feet. 



2. The spaces run through 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. 



3. Water forced out of a larger channel through one or more smaller pas- 

 sages, will have the streams through those passages contracted in the ratio of 

 25 to 21. This is shown in the 36th prop, of the 2d book of Newton's Prin- 

 cipia. 



4. 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. 



This is evident from the nature of motion. 



VOL. XI. C c 



