TRACT 38.

PROBLEMS.

367

{12 +z). y/{x — z).x — I2x~*x — Ga.x^x— (|a : + &c ;

• i _a 3

mz = + AWBl’ 4 * + ^mcx 7, X &c.

Then, equating the like terms, &c, we have

. _ 8 24 96 64 ,0

m ’ •»* ’ 5m3 ’ 3m 4 J ’

rr 8 A 24 , 96 5 , 64 , „

Hence z = —z *-. v 2 4- x* 4- —x s 8cc.

m m* 5 m3 1 3 m 4

Or z =

8 A ,

— x 1 nearly.

7?X J

But, by the first process, when x ~ 3, Z = - 886 ; whichsubstituted for them, we have z = -886, and the series =1‘63 ; therefore the correct fluents are

•886 =

l - 63 + — x^ — ^-x z &c,

m in* *

or z + *744 = — .r'*- -rx z &c.

m m?

And when z = 3 = ac, it gives x = 6’369 for the heightof the tide without, when the ditches are filled to the top ofthe sluice, or 3 feet high ; which answers to 3 h 1l' 4".

Lastty, to find the time of rising the remaining 3 feet abovethe top of the sluice ; let,v = cothe height of the tide above CD,

2 = ce ditto in the ditches above cd ;and the other dimensions as before.

Then g : </eg : : 2 g : 2 </g(x — z) ~ the ve-locity with which the water runs through thewhole sluice ad ; conseq. ad x 2 */g(x — z) =

1 Sy'’ g[x — z) is the quantity per second running through the

sluice, and V(x — z) ~v the velocity of z, or the rise

of the water in the ditches, per second; hence v : z :: l" :

i — — = x - = 1800*, and viz = xVlx — z) is

the fluxional equation; where m = —-—■ = ——

1 ’ 18 OVg 2079

To find the fluent,

A 4 X s.

Assume z = Az ,1 + B.r i -|-c;r 1 -l-D.r 2 See.

A 4 x

i hen x — £ — x — ax z — bx € — cx z See*