22$

THEORY AND PRACTICE

TRACT 37.

;part nearly in one place, viz, when the velocity is about1600 or 1700 feet per second; wc may find a more exactformula in the following manner. Let us suppose the re-sistance to be proportional to three different powers of thevelocity, that, is, partly as the square and partly as the

first power v, and parti) 7 as the \ power v i , with indetermi-

nate coefficients, or expressed by xv 1 + yv r + zv = r, or{xv -|- y*/v + z)v~r; then the values of the three assumedgeneral quantities x,y, z, will be determined, by assumingr equal to the true or experimented resistance at three seve-ral parts of the series, as at the three velocities 600, 1200,1800 feet, by which means the resulting formula will bequite correct for all these three velocities, and very nearlyso for all the other or intermediate ones. Now, the experi-mented resistances for those three velocities are 6'9, andS5 275, and 85-537; these numbers then being .substitutedseverally for r, there results the three following equations,

3

600^+ 600 T y4 COO Z = 6-9,

I200 1 -*- + 1200 * y -f 1200 s = 35-275,

1800\r -f lSOO^y + lSOOz = 85\537.

Then, dividing the first of these by COO, the second by 1200,and the third by 1800, there result the three600x + y*/ 600 4 z = "0115,

12C0j: 4 1200 + Z = -029396,

1800.r + 3 /A/ISOO + z =: 0475205.

Now, subtracting the first of these from the 2nd, and the 2ndfrom the 3rd, there result these two,

600.r + (y' 1200 - y' 60 °)^ = ’017896,

GOO.r + ( 1 / 1 800 — v /1200) y = -0181245.

And the difference of these two is

(V1800 - 2^1 -’00 4 v/600)?y = -0002285, or div. by

*/600 it is ( v /3-2v / 24v / l)'/=’00000933, and then•0(1000933 .nnnnnov*

r-

•00000933

T = Z 090376 = —’0000968, the value of y.

Hence GOO.r = -017896-(/1200 - y- 600)j/=-01887S,and also 600.r=-0l81245-(v / 1800- v / l200)y=-0l8878,

therefore x:

•018R78

600

= -000031463, the value of x.