and to act vertically upwards through the center of gravity of the fluid displaced. From these two assertions, then, we obtain that: the weight of the fluid displaced equals the weight of the floating body, and that the centers of gravity of the two lie in the same vertical line.

21. If a floating body be disturbed in its position in such a way that the amount of fluid displaced by it remains the same, the forces acting upon it will be unaltered as regards magnitude and direction, for they will be, its own weight acting vertically downwards at its center of gravity and the weight of the displaced fluid, which is, as before, equal to the weight of the body, and acts vertically upwards through its center of gravity: but in the general case these two centers of gravity will be no longer in the same vertical line, and thus a couple will have been produced, under whose action the body will either return to its original position of equilibrium or will be removed further from it, according as the new direction of the resultant of the fluid pressures, i. e. the weight of the fluid displaced, meets that fixed line in the body, which passes through its center of gravity and was vertical in the body's floating position, above or below the center of gravity. This

g' that of the fluid displaced, W the weight of the body, R the resultant of the fluid pressures on its surface and therefore equal W, M the point where the direction of R meets the fixed line through g,—the dotted figures refer to the original position of equilibrium.

The original position of the floating body is said to be one of stable equilibrium, when upon a very slight disturbance of this kind the couple produced tends to bring the body back again, and unstable when the contrary is the case: instances of these two are given in the figure. The equilibrium is said to be neutral whenever this very small displacement fails to produce a couple, i. e. when the two centers of gravity are still brought by it into the same vertical line.

It is not difficult to see that when a body floats with its center of gravity below that of the fluid displaced, the equilibrium will be stable.

Ex. In illustration of this article, consider the case of a floating body, of which the portion immersed is part of a sphere.

The direction of the fluid pressures being normal will at every point pass through the center of this spherical surface.

Therefore the direction of the resultant of the fluid pressures must, as well in the disturbed as in the floating position, pass vertically through this center.

Hence, clearly, equilibrium will be stable or unstable according as the center of gravity of the body falls below or above the center of the spherical surface.

22. If a body be immersed in a fluid of less specific gravity than itself it will sink.

Let V be the volume of such a body Q; S, S' the specific gravities of the fluid and body respectively; then the forces

=

acting upon are its own weight = VS acting vertically downwards through its center of gravity, and the resultant of the fluid pressures upon its surface; but this resultant is equal to the weight of the fluid displaced by Q, VS, and acts vertically upwards through the center of gravity of the fluid displaced, which is also that of the body, if we suppose the body and fluid each to be of uniform density; therefore on the whole the body is acted upon by a vertical force equal to the difference between these two, i. e. of the weight of the body and that of the fluid displaced by it, V(SS) in a downward direction; it must therefore sink.

=

23. If it be required to find the force to be applied by means of a string in order to hold the body in its position, it must evidently be equal and opposite to this force V (S' - S).

But the force requisite to support a heavy body or to keep it from falling under the action of gravity is taken as the measure of its weight. Hence the foregoing shews that the apparent weight of a body when immersed in a fluid is less than its real weight by the weight of the fluid displaced. This result is very useful in finding the specific gravities

of bodies.

24. If on the contrary the specific gravity of the body immersed be less than that of the fluid, it will rise: for, as before, the resultant force upon the body is a single vertical force passing through its center of gravity, equal to the difference between the weight of the body and that of the fluid displaced by it, and acting in the direction of the larger

force, which in this case is upwards. This will serve to explain the ascent of a balloon.

25. If the specific gravities of the body and fluid be the same, i. e. if S and S' be equal, this resultant force clearly vanishes, and hence the body would rest in any position of total immersion.

The remainder of this section gives some methods of comparing the specific gravities of different substances whether solid or fluid, and describes instruments called Hydrometers, which are used for the purpose: it may be remarked that in all cases the ratio between the weights of equal volumes of the two substances is the quantity sought. (Art. 11.)

26. An ordinary balance adapted to weighing bodies in fluids is sometimes termed an Hydrostatic balance: one of the

scales is small and hung very short; at the bottom of the pan is a hook from which the body, while immersed in a fluid

contained in any vessel below it, may be suspended by a small string or wire.

27. To find the specific gravity of a solid body, that of distilled water being taken as the unit.

(1) When the specific gravity of the body is greater than that of the distilled water, or in other words, when the body sinks upon immersion :

Let the weight of the body in vacuum be determined = W suppose; then let its apparent weight in distilled water be ascertained by the hydrostatic balance, suppose it= W'; then W- W' is (Art. 23) the weight of the distilled water displaced by it; if then S be the specific gravity required, (Art. 11),

S=

weight of the body

weight of an equal volume of distilled water

(2) When the specific gravity of the body is less than that of the distilled water:

Let a piece of some heavy substance be attached to the body, such that the whole will sink upon immersion; let w be the ascertained weight of this attached portion in vacuum, w' in the water, W, the weight of the compound body in vacuum, W,' the weight of the same in the water, and W, as before, the weight of the body itself in vacuum; then

1

the weight of water displaced by the compound body
when immersed W1- W

=

1

of that displaced by the attached body = w — w' ;