A. SAB. SB2 + C. SC

A. Sa + B. Sb + C. Sc by prop. 41 the distance of the centre of oscillation O, below the axis of suspension; where any of the products A. Sa, B. Sb, must be negative, when a, b, &c. lie on the other side of S; which is the same expression as that for the distance of the centre of percussion, found in prop. 46.

Hence it appears, that the centres of percussion and of oscillation are in the very same point. And therefore the properties in all the corollaries there found for the former, are to be here understood of the latter.

249. Corol. 1. If p be any particle of a body b, and d its distance from the axis of motion S; also, G, O, the centres of gravity and oscillation. Then the distance of the centre of oscillation of the body, from the axis of motion, is sum of all the pd3 SO= SG the body b

250. Corol. 2. If b denote the matter in any compound body, whose centres of gravity and oscillation are G and O; the body P, which being placed at P, where the force acts as in the last proposition, and which receives the same motion from that force as the compound body b, is P =

SG. SO
SP

b.

A SA + B. SB3 + C. SC2.

SP

For, by corol. 2, prop. 46, this body P = But, by corol. 1, prop. 45, SG. SO. b = A. SA + B. SB + C. SC2; therefore P=

SG. SO

SP

b.

251. SCHOLIUM.-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.

252. 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 shall the dis140850 tance of the centre of oscillation, be SO = inches. For, the length n n 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,

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.

253. 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:

PROP. XLIX.

254. 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 f acting at P, as the whole system receives.

Now, by cor 3, prop. 46, the angular velocity generated in the system, by the force f, is as

f. SP

A . SA2 + B. SB2, &c.; and, by the same, the angular velocity of the system placed in K, is f. SP

(A+B+C, &c.). SR; then, by making these two expressions equal to each other, we have

SR = √

A. SA + B. SB+ C. SC
A+B+C

S

B

R

for the distance of the centre of gyration below the axis of motion.

255. Corol. 1. Because A. SA' + B. SB, &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 SR SG. SO; that is, the distance of the centre of gyration, is a mean proportional between those of gravity and oscillation.

256. Corol. 2. If p denote any particle of a body b, at d distance from the sum of all the pd

axis of motion; then SR2 =

body b.

PROP. L.

257. To determine the velocity with which a ball moves, which being shot against a ballistic pendulum, causes it to vibrate through a given angle.

THE Ballistic Pendulum is a heavy block of wood MN, suspended vertically by a strong horizontal iron axis at K, to which it is connected by a firm iron stem. This problem is the application of the last proposition, or of prop. 46, 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 only method yet known for determining them with any degree of accuracy.

Let G, S, O be the centres of gravity, gyration, and oscillation, as determined by the foregoing propositions; and let P 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.

Put p = the whole weight of the pendulum,

b the weight of the ball,

=

KG the dist. of the centre of gravity,

KO the dist. of the centre of oscillation,

r = KS = go the dist. of the centre of gyration,

i = KP the distance of the point of impact,

v = the velocity of the ball,

u

the velocity of the point of impact P,

c = chord of the arc described by the point O.

By prop. 48, if the mass p be placed all at S, the pendulum will receive the

same motion from the blow in the point P; and as KP": KS* :: p:

KS2
KP2

por

go
ii

p, (prop. 46) the mass which being placed at P, the pendulum will still receive the same motion as before. Here then are two quantities of matter, go namely, b and p, the former moving with the velocity v, and striking the ii 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
гг

p+bb:: vu, and there

bii+gop fore v = bii point P, and the known dimensions and weights of the bodies.

u the velocity of the ball in terms of u, the velocity of the

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: cc: the versed sine of the arc described by O. Then by the laws 20'

сс

of falling bodies, /16 1/32 2, :

CC
20

the velocity acquired by the point O in descending through the arc whose chord is c, where a = 162 feet: and therefore o:i::c

2a ci 2a

which is the velocity u, of the point P.

Then, by substituting this value for u, the velocity of the ball, before found,

bio Xc So that the velocity of the ball, is directly as the chord of the arc described by the pendulum in its vibration. 258. SCHOLIUM.-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.

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

Therefore 656.56 × 2·1337, or 1401 feet, is the velocity, per second, with which the ball moved, when it struck the pendulum.

OF HYDROSTATICS.

260. HYDROSTATICS is the science which treats of the pressure, or weight, and equilibrium of water, and other fluids, especially those that are non-elastic, 261. A fluid is elastic, when it can be reduced into a less bulk by compression, and which restores itself to its former bulk again when the pressure is removed; as, air. And it is non-elastic, when it is not compressible or expansible; as, water, &c.

PROP. LI.

262. 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.

263. Corol. 1. Hence, water which 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.

264. 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 as the sea in respect of the earth.

PROP. LII.

265. 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 fluids.

FOR, upon every equal part of the 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.

266. 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.

267. Corol. 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.

PROP. LIII.

268. When a fluid is pressed by its own weight, or by any other force; at any point it presses equally, in all directions whatever.

THIS arises from the nature of fluidity, by which it yields to any force in any direction. If it cannot recede from any force applied, it will press against other parts of the fluid in the direction of that force. And the pressure in all directions will be the same. For if it were less in any part, the fluid would move that way, till the pressure were equal every way.

269. Corol. 1. In a vessel containing a fluid; the pressure is the same against the bottom, as against the sides, or even upwards, at the same depth.

270. Corol. 2. Hence, and from the last proposition, if ABCD be a vessel of water, and there be taken, in the base produced, DE to represent the pressure at the bottom; joining AE, and drawing any parallels to the base, as FG, HI; then shall FG represent the pressure at the depth AG, and HI the pressure at the depth AI, and so on; because the parallels, FG, HI, ED, by sim. triangles, are as the depths, AG, AI, AD; which are as the pressures, by the proposition.

F

H

E

D

B

And hence the sum of all the FG, HI, &c. or area of the triangle ADE, is as the pressure against all the points G, I, &c. that is, against the line AD. But as every point in the line CD is pressed with a force as DE, and that thence the pressure on the whole line CD is as the rectangle ED. DC, while that against the side is as the triangle ADE or AD. DE; therefore the pressure on the horizontal line DC, is to the pressure against the vertical line DA, as DC to DA. And hence, if the vessel be an upright rectangular one, the pressure on the bottom, or whole weight of the fluid, is to the pressure against