attached. Putting, therefore, gdt for u, and dv' for v', in the value of found above, N being considered as very small or nothing compared with L, the mass of the vessel, which gives

Let do" be the velocity with which that point of the vessel turns which is distant one foot from the centre of gravity; we shall have

do: do:: AG: 1 :: D: 1,

Let z be the arc described by the point in question during the time; we shall have

Therefore, if the rope acting always perpendicularly to the length of the vessel, be attached to another point I, and we call the arc described by the same point during the same time t, we shall have

280.

Geom. 294.

calling D' the distance IG; whence we have

and hence

z: z' :: D: D' :: AG: IG;

zzz :: AG IG or AI: AG.

Now if after each experiment we measure, as may easily be done in several ways, the angles of rotation, that is, the number of degrees contained in the arcs z, z respectively, we may substitute these numbers instead of the arcs z, z', in the proportion; and since the distance AI is known, we readily obtain AG, that is, the position of the centre of gravity.

The value of AG or D being determined, we calculate the length of the arc z which has 1 for radius, and of which the number of degrees is known; then, since N is known, and g is 271. equal to 32,2 feet; if we take care to observe the number of seconds which elapse up to the instant at which the number of degrees in z is counted, we shall know every thing except fm r3 in the equation

Fig.188.

whence we obtain the value of fm r2, which it would be very troublesome to obtain by a particular calculation of the differ ent parts of the vessel.

382. When a body L of any figure whatever, having received an impulse in a direction HZ, not passing through the centre of gravity, takes the two motions of which we have spoken, it is easy to see, that for an instant it may be regarded as having but one single motion, namely, a motion of rotation about a fixed point or axis F, which according to the figure of the body, and also the distance GZ at which the impulse passes from G, may be situated either within or without the body. For if, while the

line GZ is carried parallel to itself from GZ to G'Z', we imagine it to turn about the moveable point G, since the points of the body have velocities of rotation greater in proportion to their distances from G, it is manifest that there is upon the line ZG a point F which will be found to have described from F' toward F, an arc equal to GG', and which may be regarded for an instant as a straight line; the point F then will have retrograded as far by its motion of rotation as it has advanced by the velocity common to all parts of the body; this point will therefore have remained constantly in F, which, for this reason, may be considered for an instant, as a fixed point about which the body turns. If we would know the position of the point F, it will be remarked that the arcs FF', Z'I, which the points F' and Z' describe in an instant, may be considered as straight lines perpendicular to GZ, or parallel to GG'; now the similar triangles FF'G', G'Z'I, give

383. The point F is called the centre of spontaneous rotation, because it is a centre which the body takes as it were of itself. This point is precisely the centre of oscillation which the body L would have, if it turned about a fixed point or axis situated in Z; for from

361.

136.

356.

Now fmr2 + L × GZ is in article 360 precisely what we have understood by fm r2 in article 361; therefore the point F is here the same as the point O in article 361.

We perceive, therefore, that the point about which a body may be considered as turning for an instant, is independent of the value of the force or forces which are applied to this body; and generally it may be inferred from the value of FG, that this point is the more distant, according as the force in question, or the resultant of all the forces, acts at a less distance from the centre of gravity.

384. We have seen that when a body turns about a fixed point or axis, its centre of percussion is the same as its centre of oscillation; whence these two centres are found by the same operation. It is not the same when the body is free. For, let us suppose a body whose mass is L, to turn about its centre of gravity with a velocity, which, for a point situated at the known distance a, shall be v; and that at the same time this centre moves with the velocity u. It is manifest, in the first place, that the resul ing force of all the motions belonging to the different parts of this body, will have for its value L × u or L u, that is, the same as if the body had no motion of rotation. In the second place, the distance at which the resultant must pass from the centre of gravity, is evidently that at which a force equal to Lu, would produce in the body a velocity of rotation equal to that which it actually has; but this velocity v has for its expression u x D x a calling D the distance sought; we have, therefore,

Lu

f m ra

and hence we see that the distance of the centre of percussion of a free body depends on the ratio of the velocity of rotation to the velocity of the centre of gravity; and particularly that it is nothing when the velocity of rotation is nothing, as in fact it ought to be.

We may hence determine at what point to place an obstacle in order to stop a free body which has a progressive and rotatory motion at the same time; namely, at the centre of percussion of this body, or the point where it would give the strongest blow or exert the greatest force.

Method of estimating the Forces applied to Machines.

385. Any force has for its measure, as we have already said, the product of a determinate mass, into the velocity which the force in question is capable of giving to this mass. It seems proper, in this place, to add something by way of illustrating the application of this principle to machines.

When two weights act against each other by means of a simple fixed pulley, it is necessary in order to an equilibrium that their masses should be equal; and this equilibrium once established, will always remain.

But if instead of opposing a weight to a weight, we oppose the force of an animal, as that of a man, for example, although it be true that, in order to an equilibrium, this man has only to exert an effort equal to the weight to be sustained, that is, equal to the quantity of motion represented by the mass of this body multiplied into the velocity which gravity communicates in an instant; it is, nevertheless, evident that if the man were capable of but one such effort, the equilibrium would continue only for an instant, because gravity renews each successive instant the action which was destroyed in the preceding.