But regarding I as a centre of moments, we shall have, from the principle of moments,

By comparing the last two equations, we have,

v" = OK.

That is, the resultant angular velocity will be equal to the diagonal of the parallelogram described on the component angular velocities as sides.

By a course of reasoning entirely similar to that employed in demonstrating the parallelopipedon of forces, we might show, that,

If a body be acted upon by three simultaneous forces, each tending to produce rotation about separate axes intersecting each other, the resultant motion will be one of rotation about the diagonal of the parallelopipedon whose adjacent edges are the component angular velocities, and the resultant angular velocity will be represented by the length of this diagonal.

The principles just deduced are called, respectively, the parallelogram and the parallelopipedon of rotations.

Application to the Gyroscope.

152. The gyroscope is an instrument used to illustrate the laws of rotary motion. It consists essentially of a heavy wheel A, mounted upon an axle BC. This axle is attached, by means of pivots, to the inner edge of a circular hoop DE, within which the wheel A can turn freely. On

F

E

G

Fig. 132.

one side of the hoop, and in the prolongation of the axle BC, is a bar EF, having a conical hole drilled on its lower

face to receive the pointed summit of a vertical standard G. If a string be wrapped several times around the axle BC, and then rapidly unwound, so as to impart a rapid motion of rotation to the wheel A, in the direction indicated by the arrow-head, it is observed that the machine, instead of sinking downwards under the action of gravity, takes up a retrogade orbital motion about the pivot G, as indicated by the arrow-head H. For a time, the orbital motion in• creases, and, under certain circumstances, the bar EF is observed to rise upwards in a retrograde spiral direction; and, if the cavity for receiving the pivot is pretty shallow, the bar may even be thrown off the vertical standard. Instead of a bar EF, the instrument may simply have an ear at E, and be suspended from a point above by means of a string attached to the ear. The phenomena observed are the same as before.

Before explaining these phenomena, it will be necessary to point out the conventional rules for attributing proper signs to the different rotations.

GO

Fig. 133.

Let OX, OY, and OZ, be three rectangular axes. It has been agreed to call all distanges, estimated from 0, towards either X, Y, or Z, positive; consequently, all distances estimated in a contrary direction must be regarded as negative. If a body revolve about either axis, or about any line through the origin, in such a manner as to appear to an eye beyond it, in the axis and looking towards the origin, to move in the same direction as the hands of a watch, that rotation is considered positive. If rotation takes place in an opposite direction, it is negative. The arrow-head A, indicates the direction of positive rotation about the axis of X. To an eye situated beyond the body, as at X, and looking towards the origin, the motion appears to be in the same direction as the motion of the hands of a watch. The arrowhead B,

indicates the direction of positive rotation about the axis of Y, and the arrow-head C, the direction of positive rotation about the axis of Z.

Suppose the axis of the wheel of the gyroscope to coincide with the axis of X, taken horizontal; let the standard be taken to coincide with the axis of Z, the axis of Y being perpendicular to them both. Let a positive rotation be communicated to the wheel by means of a string. For a very short time dt, the angular velocity may be regarded as constant. In the same time dt, the force of gravity acts to impart a motion of positive rotation to the whole instrument about the axis of Y, which may, for an instant, be regarded as constant. Denote the former angular velocity by v, and the latter by v'. Lay off in a positive direction on the axis of X, the distance OD equal to v, and, on the positive direction of the axis of Y, the distance OP equal to v', and complete the parallelogram OF. Then (Art. 151) will OF represent the direction of the resultant axis of revolution, and the distance OF will represent the resultant angular velocity, which denote by v". In moving from OD to OF, the axis takes up a positive, or retrograde orbital motion about the axis of Z. To construct the position of the resultant axis for the second instant dt, we must compound three angular velocities. Lay off on a perpendicular to OF and OZ, the angular velocity OG due to the action of gravity during the time dt, and on OZ the angular velocity in the orbit; construct a parallelopipedon on these lines, and draw its diagonal through 0. This diagonal will coincide in direction with the resultant axis for the second instant, and its length will represent the resultant angular velocity (Art. 151). For the next instant, we may proceed as before, and so on continually. Since, in each case, the diagonal is greater than either edge of the parallelopipedon, it follows that the angular velocity will continually increase, and, were there no hurtful resistances, this increase would go on indefinitely. The effect of gravity is continually exerted to depress the centre of gravity of the

instrument, whilst the effect of the orbital rotation is to elevate it. When the latter effect prevails, the axis of the gyroscope will continually rise; when the former prevails, the gyroscope will continually descend. Whether the one or the other of these conditions will be fulfilled, depends upon the angular velocity of the wheel of the gyroscope, and upon the position of the centre of gravity of the instru ment. Were the instrument counterpoised so that the centre of gravity would lie exactly over the pivot, there would be no orbital motion, neither would the instrument rise or fall. Were the centre of gravity thrown on the opposite side of the pivot from the wheel, the rotation due to gravity would be negative, that is, the orbital motion would be direct, instead of retrograde.

153. A FLUID is a body whose particles move freely amongst each other, each particle yielding to the slightest force. Fluids are of two classes: liquids, of which water is a type, and gases, or vapors, of which air and steam are types. The distinctive property of the first class is, that they are sensibly incompressible; thus, water, on being pressed by a force of 15 lbs. on each square inch of surface, only suffers a diminution of about 200'000 of its bulk. The second class comprises those which are readily compressible; thus, air and steam are easily compressed into smaller volumes, and when the pressure is removed, they expand, so as to occupy larger volumes.

Most liquids are imperfect; that is, there is more or less adherence between their particles, giving rise to viscosity. In what follows, they will be regarded as destitute of viscosity, and homogeneous. For certain purposes, fluids may also be regarded as destitute of weight, without impairing the validity of the conclusions.

.

Principle of Equal Pressures.

154. From the nature and constitution of a fluid, it follows, that each of its particles is perfectly movable in all directions. From this fact, we deduce the following fundamental law, viz.: If a fluid is in equilibrium under the action of any forces whatever, each particle of the mass is equally pressed in all directions; for, if any particle were more strongly pressed in one direction than in the others,