It is not very important to trace out the behaviour of the system shown in [Fig. 3] beyond a very brief period immediately after the weight W is applied. The point of importance is that the precession produced by the weight is very slow, and therefore that in a given interval of time the amount precessed is very small. Further, the rate of the precession depends solely upon the friction at the journals of the axis E F and not upon the weight W or the movement of the frame K except in so far as these factors affect the friction. The less the friction the less will be the rate of precession and the amount precessed in a given time. Thus by mounting the axis E F on knife edges the friction can be made so small that the precession produced by the weight W becomes immeasurable. Hence we deduce that if friction is substantially absent at the axis E F the frame K might be violently rocked on the axis N P or even set into continuous rotation without causing the axle of the wheel either to dip or to precess.
Continuing the argument, we might mount the square frame on a vertical axis and attempt to produce rotation of the wheel about the axis H J by applying a horizontal force to one side of the square frame instead of a force V on the outer ring as shown in [Fig. 1]. A similar result would be obtained. Granted an all but total absence of friction at the journals of the vertical axis H J, the precession produced about the horizontal axis E F would be immeasurably small. Thus the frame might be set into violent motion about its vertical axis without causing the axle either to rotate in a horizontal plane or to precess in a vertical one.
Fig. 4. One Degree of Freedom Lost.
Finally, if the square frame were mounted on a horizontal axis collinear with the axle B C it might obviously be rotated about this axis without affecting the system otherwise than by increasing or reducing the rubbing speed of the axle B C in its bearings.
Since pure translation of the frame in any direction cannot apply a turning moment to the system about any axis, and as rotation of the frame about any one of the three principal axes has no effect which is measurable on the orientation of the axle, it follows that, given substantial absence of friction at the axes E F and H J, the axle of the wheel will remain constantly pointing parallel with its original position, no matter how the frame K may be moved or turned about.
Fig. 5. Second Degree of Freedom Lost.
The gyroscopic system shown in [Fig. 1] has, as we have said, “three degrees of freedom,” because its wheel is free to spin about three different axes mutually at right angles. It is to be carefully noted that it can only truly be said to have three degrees of freedom so long as the inner ring and the parts inside it are not rotated on the axis E F away from the position which in [Fig. 1] they are shown as occupying relatively to the outer ring. Thus rotation of the wheel on its axle or of the whole system inside the square frame on the axis H J leaves the three axes B C, E F, H J undisturbed at right angles to each other. But rotation of the inner ring and the parts inside it on the axis E F tends to destroy one of the degrees of freedom. If, for instance, the inner ring is rotated through 90 deg., as shown in [Fig. 4], the axle B C and the axis H J will coincide in direction. In this position the wheel cannot be rotated about a horizontal axis at right angles to E F and has therefore virtually only two degrees of freedom, namely, about the axis E F and about the axis H B C J. Again, if with the inner ring in the position shown in [Fig. 4] the outer ring is turned through 90 deg. relatively to the square frame, the system assumes the configuration shown in [Fig. 5] and the wheel loses the power of rotating about a horizontal axis in the plane of the square frame.
