Chap. XVII.

THEORY OF THE MOTION OF FLUIDS.

1131

is discharged in the latter, as the quantity discharged when the vessel is kept full may bemeasured by what would be the descent of the surface, if it could descend with the velocitywith which its descent commences.

When a vessel ABCD is pierced at thebottom and the water is only maintained atthe height G II, it does not, in issuing out,always entirely fill the hole D F, but leavesa void in the middle, forming a funnelW I Q L O P, which gives place to a cir-cular jet, whose total expenditure per secondis equal to a volume of water, comprisedby the hole E F, the height IK, and theline which expresses the velocity acquiredby a fall of the height I K.

This jet only issues when the sum ofthe velocities of the water tending to re-place that of the column in the middle isless than the uniform velocity of the samecolumn.

When a horizontal orifice N O is at the extremity of acurved tube E PL M, the height I Q of the surface G II ofthe water, with regard to the diameter of the orifice, is of noconsequence, because the curved part T PVJIL of the tubebeing filled, the air which answers to the level G II cannotissue through the orifice ; but although the water be main-tained «t the level GII, its velocity will not always be ex-pressed by the root of the height I Q, because the diameterof the orifice N O, that of the valve X Y, the height I Q,and the height I K of the level of the water above the bot-tom A D of the reservoir, may be such that the water maynot be sufficient for the orifice : for whatever be the heightof the line I Q, the reservoir will not furnish more than canissue by the hole X Y; and for the jet to act fully it is ne-cessary that the square of the diameter N O of the pipe,and the root of the height I Q of the level of the reservoirabove this pipe, should be at least equal to the product of theroot of the height G A or I K of the water in the reservoirmultiplied by the square of the diameter XY ; that is tosay, these four should be reciprocally proportionate.

It is therefore not surprising that reservoirs considerablyelevated above the issue pipe frequently form jets of littleheight, and very short of the proportion which they shouldhave ; for the passage of the water on the grating beingalways much less than the circle of the conduit pipe pre-vents the tube from being always full, and keeps the waterat one height, IIS, because the grating will only furnish a certain quantity which will causethe expenditure of the orifice to be relative to VllP, and not to VT Q-

When the summit of a rectangular orifice isabove the level of the water, as MKLN, thesum of all the velocities of the films of waterwhich issue from it may be expressed by theelements of the parabolic segment F H 1 G, andthere will be a mean, O T, which being multi-plied by the height H F will give a productequal to the superficies of this segment. Asthe portion of this element will determine themean height E Q, the following method may beadopted.

Calling E F, a; E H ,b; H F, c, and the meanheight E O, x, the sum of all the velocities ofwhich the films in the height E F are capableo a /

will bfc-^-^/ «, and the sum of all the veloci-ties of the height E A will be consequently — b S* vc sum

all the velocities issuing from the orifice, which being equal to the product of the mean velocity,

Fig. 1717.

Fig. 1718.