"It is instructive to consider the same prob lem from a somewhat different standpoint. We have already determined the nature of the ap plied forces required to turn the ideal rotating flywheel (Fig. 1) about the axis A B, in a clockwise direction when viewed from A. We found that a torque must be applied which tends to urge the end of the axle above the plane of the paper in the direction C B, and the opposite end of the axle in the direction C A. It will now be proved that the reaction of the rotating flywheel, when it turns as above, about the axis A B. produces a torque which tends to urge the end of the axle above the plane of the paper in the direction C A, and the other end of the axle in the direction C B.
"Under the given conditions, the component velocities, downward through .the plane of the paper, of the particles a, b, c, d, are all being diminished; and the consequent reactions tend to turn the axle in a clockwise direction, about the line k a, when viewed from the side k. The component velocities, upward through the plane of the paper, of the particles e, f, g, k, are all being increased, and the consequent re actions tend to turn the axle in the same direc tion. It is easily seen that the reactions due to the alterations in the velocities of the particles k, 1, m, p, q, r, s, all tend to turn the axle of the flywheel in the same direction. Thus the torque due to the reaction of the rotating flywheel when turning about the axis A B, is of the character specified above.
"The precise way in which the gyroscope (Fig. 2) acts can now be readily followed. When the frame carrying the rotating flywheel a a is first supported on the pivot o, the initial tendency is for the whole to descend toward the earth, under the action of gravity. But the pivot o prevents the end b of the axle from de scending, so that an incipient rotation about a horizontal diameter commences. The reaction due to this rotation produces a torque which tends to turn the flywheel about a vertical diam eter in the direction of the arrow s. As the flywheel is free to turn in this direction, it at once commences to do so, and in so doing generates a reacting torque opposing the in cipient rotation produced by gravity. The ac tion of gravity being opposed, the rate of (in cipient) descent of the flywheel is diminished; but so long as descent continues, a torque acting in the direction of the arrow s will be and this will increase the velocity of turning, thus increasing the torque which opposes the descent of the flywheel under the action of gravity. The flywheel, finally, acquires a rota
tional velocity in the direction of the arrow s, which prodfices a reacting torque just equal and opposite to that due to the pull of gravity. If friction were entirely absent, the flywheel would then cease to descend, and would con tinue to turn at a uniform rate in the direc tion of the arrow s. In this process, the work performed is that due to the incipient descent of the flywheel ; this work is just sufficient to supply the kinetic energy due to the rotation of the flywheel and its supporting framework about the axis o. When the permanent condi tion outlined above has been attained, no further work is done in the absence of friction. If there is friction between the supporting lug n, and the pivot o, the gyroscope will slowly descend, at such a rate that the work per formed by gravity is just equal to that needed to overcome the frictional drag.
*In the absence of friction, it is obvious that" the gyroscope turns about o as centre merely: by virtue of its own inertia, after the final state has been reached; in this respect the motion resembles that of a planet around the sun. The torque due to gravity, though neces sary, only serves the purpose of neutralizing the reacting torque which the turning of the fly wheel about a vertical diameter produces.* From the foregoing it is seen that gyroscopic action depends upon four dominant factors : (1) the moment of inertia of the rotating flywheel — that is, its diameter multiplied by its weight as distributed relatively to its axis of rotation; (2) its freedom to incline its axis of rotation in two directions; (3) the velocity with which the axis is thus inclined; (4) the velocity with which the flywheel is rotating.* Mr. C. M. Brownall also, in a treatise on The Gyroscope, an Explanation without Math ematics,' published in the Scientific American Supplement of 10 Aug. 1907, summarizes the action of the gyroscope force as follows: "I. The gyroscopic force always acts at right angles to the plane of motion of the axis, neither accelerating nor retarding it and only tending to change its direction. The gyroscopic force is of the nature of a couple, and can only be balanced by an equal couple.