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TRACT 38.

PROBLEMS.

86 i

Again, if the vessel be a ditch, or canal, of 20 feet broadat the bottom, 22 at the top, 9 deep, and 1000 feet long;then is 90 : 90 + x :: 20 : - ° ^ x x 20 the breadth of thesurface of the water when its depth in the canal is x_ ; andtherefore a = - x 20000 is the surface at that time. Con-sequently t or - ~ - x - = J000O x 9 ° + ■ * x —^r is the fluxion

6 a y x 90 3 ayr

of the time ; the correct fluent of which, when x — 0, is

10000 180 + |rf

x Vd ■

10000 x 13C x 3

20666" nearly, or

3 90 o ' 90 X 3

5 h 44' 26", the whole time of exhausting by a sluice of 1 footsquare..

PROBLEM XXVIII.

To determine the Time of emptying any Ditch, or Inundation ,Sic, by a Cut or Notch, from the 'Top to the Bottom of it.

Let x

AB,

the variable height of the descend-

ing water at any time;

b = ac, the breadth of the cut;d — the whole or first deptli of water;a = the area of the surface of the water in the ~ditch ;

g = 16 -r 1 - feet, the descent by gravity in l".

Now, the velocity at any point d, is as y bd, that is asthe ordinate de of a parabola, bec, whose base is ac, and al-titude ab. Therefore the velocities at all the points in ab,are as all the ordinates of the parabola. Consequently, thequantity of water running through the cut abgc, in anytime, is to the quantity which would run through an equalaperture placed all at the bottom, in the same time, as thearea of the parabola ABC, to the area of the parallelogramabgc, that is, as 2 to 3. .

But f g : f x : : 2 g : 2 >Zgx, the velocity at ac ; there-fore 2 fgx x b.v x -| = } bxVgx is the quantity discharged

per second through abgc; and consequently — iN . F i s thevelocity per second of the descending surface. Hence then