OF HYDRAULICS. 



61 



pirticles renders »hem more easily actuated by the pressure 

 of the succeeding column. Still, however, some deduction 

 must be made for the lateral motions of the neighbouring 

 particles, which tend rather to diminish the quantity of the 

 discharge, than to lessen the actual velocity of the jet: the 

 particles approaching and even passing through the orifice 

 obliquely, contract the diameter of the stream nearly in the 

 ratio of 4 to 5, when the aperture is in a thin plate ; but the 

 velocity in the contracted part is only one fortieth or one 

 fiftieth less thart»that which is due to the height. 



Scholium. The velocity of the discharge through dif- 

 ferent kinds of apertures may be found by multiplying the 

 square root of the height in feet by 'a certain coefficient; 

 this, for the undiminished velocity, is 8.0229 ; for an orifice 

 imitating the form of the contracted stream 7.8 ; for bridges 

 with pointed piers 7.7 ; for bridges with square piers O.y ; 

 for short pipes, from two to four times as long as their dia- 

 meter, 6.6 ; for orifices in a thin plate, and for weres, about 

 i. When the orifice is made between two reservoirs, the 

 discharge is nearly in the same relation to the ditference 

 of their heights. 



383. Theorem. A jet of water issuing 

 from aa orifice of a proper form, and directed 

 upwards, rises nearly to the lieight of the 

 head of water in the reservoir. 



For it has been shown, that the velocity is nearly equal to 

 that which is produced by the fall of a body through the 

 height, and each of the particles may be considered nearly 

 its a separate projectile. 



384. Theorem. If a jet issue horizon- 

 tally from any part of the side of a vessel 

 standing on a horizontal plane, and a circle 

 be described having the whole height of the 

 fluid for its diameter, the fluid will reach the 

 plane at a distance from the vessel, eqnal to 

 that chord of the circle in which the jet ini- 

 tially moves. 



The horizontal velocity ot the 

 jet, being equal to that which is 

 acquired by a body falling through 

 the distance AB below the sur- 

 face, would describe in the time 

 of falling through AB, a distance 

 equal to 2AB (233), and in the time of falling through BC, 

 in which the jet will reach the horizontal plane (25l), a 

 distance greater in the ratio of those times, or of the square 

 roots of the spaces (235)- Call AC, 1, then (121) 1 : AD ;: 



AD ! AB, ADq=:AB, and AD=v^ AB ; in the same man- 

 ner CD" v'BC, therefore the times are as AD and CD: 

 but AD : CD : : AB : BD, and 2BD, or DE will be equal to 

 the space CF described by the horizontal velocity (251} in 

 the time of falling through BC. 



385. Theorem. When a cylindrical or 

 prismatic vessel empties itself by a small ori- 

 fice, the velocity at the surface is uniformly 

 retarded; and in the lime which it occupies 

 in eiilptying itself, twice the quantity would 

 be discharged if it were kept full by a new 

 supply. 



For the velocity of the surface is in a constant ratio to 

 that at the orifice; and since the velocity varies as the 

 square root of the height, or of the space to be described, 

 the law of the motion will be the same as in the ascent of 

 a projectile (230, 233, 236), and the space described by 

 every such motion is half the space that would be described 

 by the initial velocity. 



386. Theorem, The quantity of a fluid • 

 discharged through an aperture of equal 

 breadth, continued from the bottom of a re- 

 servoir to the surface, is two thirds of that 

 which would be discharged with the velocity 



at the bottom. ^ 



For the velocity at the distance x below the surface is 



Os/r, and the fluxion of the discharge a^/xi or ax'i, of 

 which the fluent is § oa'x, which is |of what would be dis- 

 charged with the velocity ov'.r. 



387. Theorem. The friction of 'fluids 

 varies nearly as the square of the velocity. 



The friction appear? to depend principally on the centri- 

 fugal force of the particles of the fluid moving in certain 

 curves in order to surmount minute obstacles; hence it 

 may be compared with the force of a revolving body, which 

 always varies as the square of the velocity. The identity 

 of the curves in all cases could however scarcely be inferred 

 from theory only, if it were not suppor«d by experience. 

 The viscidity of the fluid seldom adds much to the resist- 

 ance. 



388. Definition. The hydraulic mean 

 depth of a river is the quotient of the area of 

 its section, divided by the length of the out- 

 line of the section in contact with the bot- 

 tom. 



