WATER. 



an invariable height, and determines the time required, by- 

 comparing it with that of a veflel emptying itfelf by the 

 preffure of the water that it contains, obferving, that the 

 motion is retarded in both cafes in a fimilar manner, and 

 he finds the calculation agree fufBciently well with expe- 

 riments made on a large fcale. 



Rules for meafur'wg the Quantity of Water which fows over 

 a Weir, or through an aperture in the Edge of a Board, the Stream 

 being open atTop. — If we fuppofe water running in a regular 

 fheet over the edge of a large ciftern or refervoir, or through 

 a reftangular aperture made in the perpendicular wall or 

 fide of the ciftern, but open at top, we may take the area 

 of the aperture, and proceed to find the velocity by calcu- 

 lation. 



When this fubjeft has been confidered theoretically, 

 it has been affumed, that the furface of the water at 

 the place where it runs through the aperture, is with- 

 out motion, becaufe it ftands at the fame level with the 

 ftagnant water in the refervoir, and that the velocity 

 of the water at different depths will always be as the 

 fquare root of the depth ; that is, beginning at nothing at 

 the furface, the velocity at different depths will increafe by 

 that law. 



We can find the velocity at the bottom of the aperture, or 

 at any intermediate depth, by the rules and table we have 

 already given ; but what we require is the mean velocity of 

 the whole (heet of water. We could obtain this nearly by cal- 

 culating the velocities for a great number of different depths, 

 increafing by regular intervals, and taking a mean of the 

 whole ; but we can effeft the fame with exaftnefs, if we take 

 two-thirds of the velocity at the bottom, and confider it as 

 the mean velocity of the whole body of water ; or, the ve- 

 locity due to four-ninths of the depth, will give the fame 

 refult. 



In praftice we muft make allowance for lofs of motion by the 

 friftion of the water in pafling through the aperture, and alfo 

 becaufe the water does not fill the aperture to the fame level 

 as the ftagnant water in the refervoir. The motion of the 

 water extends fome diftance into the refervoir, and the water 

 will confequently have a floping furface from that part of 

 the furface where the motion begins ; the flope will con- 

 tinually increafe as the motion of the water accelerates, fo 

 as to form a convex furface, wliich is a portion of a para- 

 bolic curve ; hence the furface of the water where it is 

 pafling through the aperture will be in rapid motion, infleadof 

 being raotionlefs as the theory fuppofes, and the furface will 

 be much lower than the furface of the ftagnant water, fo 

 that the aperture will only be half full of water ; at leaft this 

 is the affertion of M. Du Buat. But Dr. Robifon ftates, 

 that he always found the depth of the water in the aperture 

 about .715 of the whole depth from the bottom of the aper- 

 ture to the level of the water in the refervoir. 



M. Du Buat's theorem for the difcharge through an open 

 aperture, when reduced to Englifh meafures, is this : having 

 given the depth from the level furface of the water to the 

 bottom of the aperture, and alfo the width of the aperture 

 in inches, to find the difcharge in cubic inches per fecond. 



Let it be remembered that 11. 449 1 cubic inches of 

 water, or 1 1.5, will be difcharged in a fecond, through every 

 inch in width of the aperture, when the bottom of it is ex- 

 aftly one inch beneath the level furface of the refervoir. 

 To obtain the difcharge for any other depths, this number 

 muft be multiplied by the fquare root of the cube of the 

 depth in inches, and it will give the cubic inches difcharged 

 per fecond through each incli in width of the aperture. 

 Example^ — Suppofe the depth of the bottom of the aper- 



3 



ture beneath the level furface of the water in the refervoir 

 to be 4 inches. The cube of this is 64, the fquare root of 

 which is 8 ; therefore, at that depth each inch in width will 

 difcharge 8 x 11.5 = 92 cubic inches ^^r fecond ; if the 

 width of the aperture was 3 feet, then 92 x 36 inches 

 = 3312 cubic inches, or 1.917 cubic feet, which x 60 fe- 

 conds = 11.J02 cubic ktX. per minute. 



Dr. Robifon gives the following table, which is rather 

 greater than from the above theorem, and will be found 

 very exaft, when the aperture is made in a plank or 

 board half an inch or an inch thick, and fo fituated that the 

 fides and bottom of the refervoir do not corrcfpond with the 

 edge of the aperture, to lead the particles of water in a 

 current to the aperture. 



In taking the depth, if it does not exceed four inches, it 

 will not be exadf enough to take proportional parts for the 

 fraftions of an inch. The following method is exaft : if 

 there be odd quarters of an inch, look in the table for as 

 many inches as the depth contains quarters, and take the 

 eighth part of the anfwer. Thus, for 3|- inches take the 

 eighth part of 23.419, which correfponds to 15 inches. 

 This is 2.927. 



If the aperture is not in the fide of a large refervoir, but in 

 a running ftream, we muft augment the difcharge, by multi- 

 plying the feftion by the velocity of the ftream. But this 

 correftion can feldom occur in praftice, becaufe in this cafe 

 tlie difcharge is previoufly known. 



The amount of the allowance for friftion and lofs of 

 motion muft be different in different cafes, according to the 

 kind of aperture, or board over which the water flows ; but 

 will always be very nearly the fame as the allowance, for lofs 

 in an aperture or orifice of fimilar nature. For inftance, if 

 the edges of the aperture through which the water runs be 

 a thin plate, then we may find the velocity in feet per 

 minute due to the whole depth from the bottom of the 

 notch to the level furface of the water in the refervoir ; mul- 

 tiply the fquare root of the depth in inches by 85.87, as we 



have 



