GRIST AND SAW-MILLS. 



349 



of water in the JDenflock, and d the height of the gateway, 

 and is only an approximation, though very near the truth; 

 the genuine method derived from the parabola is as follows : 



Let ABCD Fig. i. reprefent a large ciftern or penflock, and 

 MKLN an orifice made in one of its fides. 



When the upper edge of the gateway, as KL is below the 

 furface of the water iu^the penfiock, the fum of all the velo- 

 cities or fiieets of water which flow through it, being exprefled 

 by the elements of the fegment of a parabola FHIG, there will 

 be found amongft them a mean ordinate OT, which being mul- 

 tiplied by the height HP, will give a product equal to the area 

 of this fegment. Now, in order to determine the mean height 

 EO, let EPr:^, EH=:/?, HPizriT, and the mean height EO=r.v. 

 The fum of all the velocities, or the area of the parabola EPG 



will be ~y/a, and the fum of all the velocities acquired by 



falling from E to H = — ^/^ ; confequently — */« — — ^ b 



will give the fum of all the velocities which flow through the 

 orifice MKLN, which is equal to the parabolic fegment HPIG, 

 or to the produft of the mean velocity \/EO («•") by the height 



HP (0, hence we have — ^a — —s/b = c^x, which equation 



being reduced will give .v = ± ^ a' +b'-2abs/'^b ^ 

 9 cc 



EXAMPLE. 



If the height EP (a) be = 8 feet, EH (h) - 6 feet, then 

 will HP or f =z 2 feet ; and by fubfiituting thefe numbers for 

 their refpeftive values in the above equation, x will be found ~ 

 6.99 feet. 



By the common praftical rule, (fee article 12,) D — ■ 



-'=b=. X, where Drr 8 and d =: 2 ; confequently h — y feeT, 



whence it appears that a — - = D — - is fufficlently exaft for 



2 2 



all common purpofes. 



In 



