TRACT 38.

PROBLEMS.

363

Let acdb represent the area of the vertical sluice, beinga square ot 9 square feet, and ab level with the bottom ofthe ditch. And suppose the ditch filled to any height ae,the surface being then at ef.

Put a = 9 the height of the head or supply;b = 3 = ab = ac ;

A = the area of a horizontal section ofthe ditches;x = a — ae, the height of the headabove ef.

Then g : x : : 2 g •. 2 V'gx the velocity with which the

water presses through the part aefb ; and theref. 2*ggxxaefb — 2bVgx ( a — x) is the quantity per second runningthrough aefb. Also, the quantity running per secondthrough ecdf is 2*/gx x 44ecdf = ^b^/gxij) — a + x)nearly. For the real quantity is, by proceeding as in the lastprob. the difference between two parab. segs. the alt. of theone being x, its base b, and the alt. of the other a — 6; andthe medium of that dif. between its greatest state at ab,where it is ad, and its least state at cd, where it is 0, isnearly Consequently the sum of the two, or \b*/gx

(a + lib — x) is the quantity per second running in by thewhole sluice acdb. Hence then ib^/gx x -- + 11 '~~’ r — v is

the rate or velocity per second with which the water rises in

_I_

the ditches : and so u : — x : : 1" : / = — — = —~ x -— -

■ v b ^ c — x

the fluxion of the time of filling to any height ae, puttingc = a + 1 lb.

Now -when the ditches are of equal width throughout, ais a constant quantity, and in that case a correct fluent of

this fluxion is t — x log. x o /’ c ~^ x thegene-

ral expression for tiie time of filling to any height ae, ora — x, not exceeding the height ac of the sluice. And■jyhen .r = ac — a — b = d suppose, then t = x log.