366

MISCELLANEOUS

TRACT 38.

• 3 6 0 1 Z

and mz = Am a x x x + \m-&x^x -f- V mcx*x -I- y nmx T i- &c.Then equating the coefficients of the like terms,so shall and consequently

4-mA = 2,

■i-mB = 0,

y?7zc. = - |a%

V 4 »l D =

&c ;

A = —,5 m

B = 0,

24

3 A 3

4 A

D =

275m3 316

S75m 4 ’

&C.

Which values of a, b, c, &c, substituted in the assumedvalue of z, give

4 5

Z = —X 15 m

24

275 m 3

ijt

X *

_16_

875m l

*4 o

x * &c :

4 V 1

or z = —x* very nearly.

5 m J

And when a: = 3 = ac, then z — '386 of a foot, or 10finches, = ae, the height of the water in the ditches whenthe tide is at cd or 3 feet high without, or in the first hourand half of time.

Again, to find the time, after the above, whenef arrives at cd, or when the water in theditches arrives as high as the top of the sluice.

The notation remaining as before,then 2 bz\/g{x — z) per sec. runs through af,and |Z>(3 — z)*/g(x — z) per sec. thro’ ed nearly ;therefore ^b^/gx (\2-\-z)</(x — z) is the whole per secondthrough ad nearly.

2

z) = v is the velocity per

conseq. —^ X (12 -\-z)i/(x

second of the point e ; and therefore

x

-- = 1800a, or

34/x/g ~ (lZ + zJ^x—x)

mz=(l2 + z)V(x—z) .a, wherem = T2o^7 J = 23 *t near] y*

3 4 S 6

Assume z = ax x + n.r^ + cx* + dx t &c. So shall

„ * A 2- a’ + 4b X

V'(r-z) = x* - -g—.r

a 3 + 4ab + 8c 4

-- X 1 &c;

3 4 5

12 + z = 12 + + bx x + cx * &c ;

16