362

MISCELLANEOUS

TRACT 38.

4 fav '— : — 'x :: l"; —— t the fluxion of the time of de-

3 a 4 bx */gx ’

scending.

Now when a the surface of the water is constant, or theditch is equally broad throughout, the correct fluent of thisfluxion gives t — X t/ d ~y x „ f or t [j e general time of sink-

° 2 b^/g b

ing the surface to any depth x. And when x = O, this ex-pression is infinite ; which shows that the time of a completeexhaustion is infinite.

But if d = 9 feet, b = 2 feet, a = 21 X 1000 = 21000,and it be required to exhaust the water down to of a footdeep; then x — T ' ? , and the above expression becomes■■ — x -—f = 14400'', or just 4 hours for that time.

And if it be required to depress it 8 feet, or till 1 foot depthof water remain in the ditch, the time of sinking the waterto that point will be 43' 38".

Again, if the ditch be the same depth and length as before,but 20 feet broad at bottom, and 22 at top ; then the de-scending surface will be a variable quantity, and, by prob.27, it will be — X 20000; hence in this case the flux, of

the time, or ———, becomes ——- x —— * ; the correct

a c i ■ i * . 1000 .90-1 90—rf, r . . c

fluent of which is t = x (■—;- -rr) i° r the time or

sinking the water to any depth x.

Now when x — 0, this expression for the complete ex-haustion becomes infinite.

But if . . x — 1 foot, the time t is 42' 56"i.

And when x = -^foot, the time is 3 h 50' 28''

PROBLEM XXIX.

To determine the Time of filling the Ditches of a Fortification6 Feet deep with Water , through the Sluice of a Trunk of3 Feet Square, the Bottom of which is level with the Bottomof the Ditch, and the Height of the supplying Water is fFeet above the Bottom of the Ditch.