ON THE REGULATION OF HYDRAULIC FORCES. 245 



we shall find that the velocity of the wheel, and consequently its hreadth, 

 and the magnitude of its buckets, is perfectly indifferent with respect to the 

 value of its operation : for supposing the stream to enter the buckets with 

 the uniform velocity of the wheel, the quantity of water in the wheel at any 

 one time, and consequently the pressure, must be inversely as the velocity, 

 so that the product of the force into the velocity will be the same, however 

 they may separately vary. If, however, the velocity were to become very 

 considerable, it would be necessary to sacrifice a material part of the fall, 

 in order that the water might acquire this velocity before its arrival at the 

 wheel ; but a fall of one foot, or even less, is sufficient for producing any 

 velocity that would be practically convenient : and it is obvious, on the 

 other hand, that a certain velocity may be procured from a wheel moving 

 rapidly, with less machinery than from another which moves more slowly. 

 In general the velocity of the surface of the wheel is between two and six 

 feet in a second ;* and whether it be greater or smaller, the force actually 

 applied will always be equal in effect to the weight of a portion of the 

 stream employed, equal in length to the height of the wheel. In order to 

 avoid the resistance which might be occasioned by the stagnant water below 

 the wheel, it is a good practice to turn the stream backwards upon its 

 nearer half, so that the water, when discharged, may run off in the general 

 direction of its motion. (Plate XXII. Fig. 290.) 



If we suffer the stream of water to acquire the utmost velocity that the 

 whole fall can produce, and to strike horizontally against the floatboards of 

 an undershot wheel, or if we wish to employ the force of a river running in 

 a direction nearly horizontal, the wheel must move, in order to produce the 

 greatest effect, with half the velocity of the stream.t For the whole quan- 

 tity of water impelling the floatboards is nearly the same, whatever may be 

 the velocity, especially if the wheel is properly inclosed in a narrow chan- 

 nel, and hence it is easy to calculate that the greatest possible effect will be 

 produced when the relative velocity of the stream, striking the floatboards, 

 is equal to the velocity of the wheel itself. The pressure on the floatboards 

 is equal to that of a stream containing the same quantity of water, and 

 striking a fixed obstacle with half the velocity, that is, such a stream as 

 escapes from the wheel, which must be twice as deep or twice as wide as 

 the original stream, since its motion is only one half as rapid ; and a column 

 of such a stream, of twice the height due to its velocity, that is, of half the 

 height of the fall, being, as we have already seen, the measure of the 

 hydraulic pressure, this force will be precisely half as great as that of a 

 similar column, acting on an overshot wheel, which moves with the same 

 velocity.;}: But the stream thus retarded will not retain the other half of 

 its mechanical power ; since its greatest effect will be in the same propor- 

 tion to that of an equal stream acting on an overshot wheel with one fourth 

 of the fall of the former : and the remaining fourth of the power is lost in 



* Smeaton, Ph. Tr. 1759, li. 134, deduces from experiments a little more than 

 three feet in a second, and observes, that high wheels (24 feet, or the like) may de- 

 viate more from this velocity than low, without materially affecting their work. 



f Do. ibid. p. 122, gives the best proportion as 2 : 5. Compare Robison, Mech. 

 Phil. ii. 625. J Ibid. p. 130. 



