292. By the method of Fluxions, the centre of oscillation, for a regular body, will be found from cor. 1. But for an irregular one; suspend it at the given point; and hang up also a simple pendulum of such a length, that making them both vibrate, they may keep time together. Then the length of the simple pendulum, is equal to the distance of the centre of oscillation of the body, below the point of suspension.

293. Or it will be still better found thus: Suspend the body very freely by the given point, and make it vibrate in small arcs, counting the number of vibrations it makes in any time, as a minute, by a good stop watch; and let that number of vibrations made in a minute be called n: Then 140850 shall the distance of the centre of oscillation, be so =

nn

inches. For, the length of the pendulum vibrating seconds, or 60 times in a minute, being 39 inches; and the lengths of pendulums being reciprocally as the square of the number of vibrations made in the same time; therefore

n2: 602 :: 39:

602 × 391/ 140850

n n

==

: the length of the pendulum which vibrates n times in a minute, or the distance of the centre of oscillation below the axis of motion.

294. The foregoing determination of the point, into which all the matter of a body being collected, it shall oscillate in the same manner as before, only respects the case in which the body is put in motion by the gravity of its own particles, and the point is the centre of oscillation: but when the body is put in motion by some other extraneous force, instead of its gravity, then the point is different from the former, and is called the Centre of Gyration; which is determined in the following manner:

PRO

PROPOSITION LVII.

295. To determine the Centre of Gyration of a Compound Body or of a System of Bodies.

LET R be the centre of gyration, or the point into which all the particles A, B, C, &c, being collected, it shall receive the same angular motion from a force facting at P, as the whole system

by the same, the angular velocity of the system placed in R,

is

f. Sp

(A + B + C &C). SR2: then, by making these two expressions equal to each other, the equation gives

SR V

A SA2 + B SBC. SC2

for the distance of the

centre of gyration below the axis of motion.

296. Corol. 1. Because A. SA2 + B SB2 &c SG. so. b, where G is the centre of gravity, o the centre of oscillation, and b the body A + B + C &c; therefore SR2 = SG. SO; that is, the distance of the centre of gyration, is a mean pro portional between those of gravity and oscillation.

297. Corol. 2. If p denote any particle of a body b, at d di

stance from the axis of motion; then SR2 =

PROPOSITION LVIII.

sum of all thepd

body b

298. To determine the Velocity with which a Ball maves, which being shot against a Ballistic Pendulum, causes it to vibrate through a given Angle.

This

THE Ballistic Pendulum is a heavy block of wood MN, suspended vertically by a strong horizontal iron axis at s, to which it is connected by a firm iron stem. problem is the application of the last proposition, or of prop. 54, and was invented by the very ingenious Mr. Robins, to determine the initial velocities of military projectiles; a circumstance very useful in that science; and it is the best method yet known for determining them with any degree of accuracy.

Let G, R, O be the centres of gravity, gyration, and oscillation, as determined by the foregoing propositions; and let be the point where the ball strikes the face of the pendulum; the momentum of which, or the product of its weight and velocity, is expressed by the force f, acting

at P, in the foregoing propositions. Now,

By prop. 56, if the mass p be placed all at R, the pendulum will receive the same motion from the blow in the SR2

go
ii

point P : and as sp2: SR2::p:orpor P, (prop. 54),

the mass which being placed at P, the pendulum will still receive the same motion as before. Here then are two

quantities of matter, namely, b and

go

is the former moving

with the velocity v, and striking the latter at rest; to determine their common velocity u, with which they will jointly proceed forward together after the stroke. In which case, by the law of the impact of non-elastic bodies, we have go bii + gop ° 1⁄2 p + b : b : : v : u, and therefore v =

bii

u the velocity of the ball in terms of u, the velocity of the point P, · and the known dimensions and weights of the bodies,

But now to determine the value of u, we must have recourse to the angle through which the pendulum vibrates; for when the pendulum descends down again to the vertical position, it will have acquired the same velocity with which it began to ascend, and, by the laws of falling bodies, the velocity of the centre of oscillation is such, as a heavy body would acquire by freely falling through the versed sine of the arc described by the same centre o. But the chord of that arc is c, and its radius is o; and, by the nature of the circle, the chord is a mean proportional between the versed sine and diameter, therefore 20 : c :: c: the versed sine of the arc described by o. Then, by the laws of falling bodies

CC

, 20

16

cc

2a

N :: 32: c/ the velocity acquired by the

20

point o in descending through the arc whose chord is c,

where a 16 feet: and therefore o:i::c

ΤΖ

which is the velocity u, of the point P.

2a ci 2a

Then, by substituting this value for u, the velocity of the

ball, before found, becomes v =

bii+gop bio

2a

× CN So

.

0

that the velocity of the ball is directly as the chord of the arc described by the pendulum in its vibration.

SCHOLIUM.

299. In the foregoing solution, the change in the centre of oscillation is omitted, which is caused by the ball lodging in the point P. But the allowance for that small change, and that of some other small quantities, may be seen in my Tracts, where all the circumstances of this method are treated at full length.

300. For an example in numbers of this method, suppose the weights and dimensions to be as follow: namely,

=7065 feet

i 94, inc. And

c18.73inc.

Therefore 656 56 x 2.1337, or 1401 feet, is the velocity, per second, with which the ball moved when it, struck the pendulum.

OF HYDROSTATICS.

301. HYDROSTATICS is the science which treats of the pressure, or weight, and equilibrium of water and other fluids, especially those that are non-elastic.

302. A fluid is elastic, when it can be reduced into a less volume by compression, and which restores itself to its former And it is bulk again when the pressure is removed; as air. non-elastic, when it is not compressible by such force; as water, &c,

PRO

PROPOSITION LIX.

303. If any Part of a Fluid be raised higher than the rest, by any Force, and then left to itself; the higher Parts will descend to the lower Places, and the Fluid will not rest, till its Surface be quite even and level.

FOR, the parts of a fluid being easily moveable every way, the higher parts will descend by their superior gravity, and raise the lower parts, till the whole come to rest in a level or horizontal plane.

304. Corol. 1. Hence, water that communicates with other water, by means of a close canal or pipe, will stand at the same height in both places. Like as water in the two legs of a syphon.

305. Corol. 2. For the same reason, if a fluid gravitate towards a centre; it will dispose itself into a spherical figure, the centre of which is the centre of force. Like the sea in respect of the earth.

PROPOSITION LX.

306. When a Fluid is at Rest in a Vessel, the Base of which is Parallel to the Horizon; Equal Parts of the Base are Equally Pressed by the Fluid.

FOR, on every equal part of this base there is an equal column of the fluid supported by it. And as all the columns are of equal height, by the last proposition they are of equal weight, and therefore they press the base equally; that is, equal parts of the base sustain an equal pressure.

307. Corol. 1. All parts of the fluid press equally at the same depth. For, if a plane parallel to the horizon be conceived to be drawn at that depth; then the pressure being the same in any part of that plane, by the proposition, therefore the parts of the fluid, instead of the plane, sustain the same pressure at the same depth. ·

308. Cord. 2. The pressure of the fluid at any depth, is as the depth of the fluid. For the pressure is as the weight, and the weight is as the height of the fluid.

309. Corol