368

MISCELLANEOUS

TRACT 38.

. . , ■, A . A * . A 2 + 4b 3. . -

X»/(X — Z)= X*x - ttX^X — •—-—-I 1 * &C.

M O

MJ2 = InAX^X + £nBX*X + ^nc-x*x Sec.

Then equating the like terms gives

2 _ -1 _ 1 -1 Sr' 3 D _ 810« 4 ’ 8cC "

”3«'

B =

Hence z = ~-x *3m

6» 25

1

907i 3 •

, , 1 i.

— X" -I- X 2

6m* ~ 90n 3

-x 3 See.

SlOn 4

But by tiie second case, when 2 = 0, x = 3'369, whichbeing used in the series, it is 1*936; therefore the correct

fluent is 2 = — 1-936 + —x^ — X x* &c. And when

3n On 2

Z = 3, x = 7 ; the heights above the top of the sluice,answering to 6 and 10 feet above the bottom of the ditches.That is, for the water to rise to the height of 6 feet withinthe ditches, it is necessary for the tide to rise to 10 feet with-out, which just answers to 5 hours; and so long it wouldtake to fill the ditches 6 feet deep with water, their horizon-tal area being 200000 square feet.

Further, when x = 6, then 2 = 2-117 the height abovethe top of the sluice ; to which add 3, the height of the sluice,and the sum 5-117, is the depth of water in the ditches in 4hours and a half, or when the tide has risen to the height of9 feet without the ditches.

Note. In the foregoing problems, concerning the effluxof water, it is taken for granted that the velocity is the sameas that which is due to the whole height of the surface of thesupplying water : a supposition which agrees with the prin-ciples of the greater number of authors: though some makethe velocity to be that which is due to the half height only:and others make it still less.

Also in some places, wdiere the difference between two pa-rabolic segments was to be taken, in estimating the meanvelocity of the water through a variable orifice, I have useda near mean value of the expression ; which makes the ope-,ration of finding the fluents much more easy, and is at thesame time sufficiently exact for the purpose in hand.