TRACT 38.

PROBLEMS.

259

For, the velocity at gh is to the velo-city at il, as \/eg to V 121 ; that is, asgh or il to ik, the ordinate of a para-bola ekh, whose axis is eg. Thereforethe sum of the velocities at ail the pointsi, is to as many times the velocity at g,as the sum of all the ordinates ik, to thesum of all the il’s ; namely, as the area of the parabola egh,is to the area eghf; that is, the quantity running throughthe notch eh, is to the quantity runuing through an equalhorizontal area placed at gh, as eghke, to eghf, or as 2 to3; the area of a parabola being ~ of its circumscribing pa-rallelogram.

Corol. I. The mean velocity of the water in the notch, isequal to ■§■ of that at gh.

Carol. 2. The quantity flowing through the hole ighl, isto that which would flow through an equal orifice placed aslow as gh, as the parabolic frustum ighic, is to the rectangleighl. As appears from tire demonstration.

Lr A"! ,4_

Evi p, '-A-i

PROBLEM XXVI.

To determine the Time of filling the Ditches of a Work withWater at the Top, bi/ a Sluice of 2 Feet square; the Head,of Water above the Sluice being 10 Feet, and the Dimensionsof the Ditch being 20 Feet wide at Bottom, 22 at Top, 9deep, and 1000 Feet long.

The capacity of the ditch is 189000 cubic feet.

But vT6 : \/10 : : 32 : 8yT0 the velocity of the waterthrough the sluice, the area of which is 4 square feet; there-fore 10 is the quantity per second running through it;or 24 V10 only when reduced A for the contraction of thestream; consequently 24<flO : 1B9000 : : ^ == 249C"

or 41' 10" nearly, is the time of filling the ditch.