It may be said, perhaps, that we have neglected to discuss what happens if from the position I the wheel is turned by some agency right round until the end B of the axle points due south. In this condition there is no resultant gravity torque on the pendulum axis, and the axle is lying parallel with the earth’s polar axis, so that the rotation of the earth does not cause relative motion between the wheel, etc., and the frame. Just as in the original configuration I, there is thus in this condition no force applied to the system tending to make the end B swing away from the pole. But as the reader may readily establish for himself by reversing the arrows and reference letters in the five diagrams of [Fig. 10], the slightest departure of the end B of the axle from the south-pointing direction towards either side of the meridian will call into play a force which will cause the end B to precess up towards the north. With the wheel spinning in the direction we have shown throughout our illustrations the only stable position of equilibrium for the axle is the north and south with the end B pointing north. It may be pointed out that the magnetic needle can, like the gyro-compass axle, assume a position of unstable equilibrium with the north-seeking end pointing south.
A point of very great practical importance into which to inquire is the magnitude of the directive force, the existence of which, when the axle is deflected from the north and south position, we have just demonstrated. This directive force or restoring moment, as will have been gathered from our explanation, increases with the deflection from the north, being a maximum when the axle is lying east and west or west and east. Its magnitude in any position of the axle depends upon (a) the speed of rotation of the earth on its polar axis, (b) the speed of the spinning wheel on its axle, and (c) the mass, or, more correctly, the moment of inertia, of the wheel. The first item is small—0.0007 of a revolution per minute—and is quite beyond our control. The second factor is consequently made as large as possible, while the third is also made large, but a limit is placed to our choice by questions of safety and temperature rise at the high speeds adopted for the spin of the wheel. In the following table we give the values of these factors for three of the types of gyro-compass to be described later.
| Compass | Wheel diam. in. | Wheel weight lbs. | Speed r.p.m. |
| Anschütz[1] | 6 | 10 | 20,000 |
| Sperry | 12 | 45 | 8,600 |
| Brown | 4 | 4¼ | 15,000 |
The value of the directive force for the same three gyro-compasses and for an ordinary magnetic compass is given in the next table, (1) for the axle—or needle—lying due east and west, and (2) for the axle—or needle—deflected 1 deg. east or west of north—true or magnetic.
Directive Force at Equator.
| Axle (or needle) E. and W. | Axle (or needle) 1 deg. E. or W. of N. | |||
| Force Grains | Leverage in. | Force Grains | Leverage in. | |
| Anschütz | 145 | 1 | 2 | 1 |
| Sperry. | 1140 | 1 | 20 | 1 |
| Brown | 12 | 1 | ⅕ | 1 |
| Magnetic[2] | 40 | 1 | ⅔ | 1 |
Fig. 11. Elementary Compass at 55 deg. N. Lat.
We have now to explain how the gyro-pendulum system manifests its compass-like property when it is transferred from the equator to some degree of latitude north or south. In [Fig. 11] we represent the system as set up in the latitude of the British Isles. The axle is horizontal and the end B is pointing due east. In this configuration the earth’s rotation is, through the action of gravity on the pendulum bob S, trying to make the wheel turn round the earth’s polar axis once every twenty-four hours. As before, in accordance with the fundamental property of the gyroscope, the wheel will try to set its axle into coincidence with or parallel with the axis about which it is being forced to rotate. In other words, the wheel will endeavour to turn in such a way as to align its axle along V U with the end B towards U. This movement can be effected by a rotation about the vertical axis H J through a right angle combined with a rotation about the horizontal axis E F through an angle θ equal to the latitude of the station at which the system is set up. The rotation about the vertical axis H J does not result in deflecting the weight S away from the plumb line, and therefore can be completely fulfilled. The rotation about the horizontal axis does, however, affect the bob. The axle, having executed the horizontal portion of its movement, is pointing its end B due north, but this end, unlike its behaviour at the equator, manifests a desire to rise vertically so as to align the axle along V U. Its desire to do so is resisted by the bob S, and the axle therefore fails to complete the full movement.
The axle is thus held substantially horizontal with its end B pointing to the north. As the earth rotates the desire of the axle to align itself parallel with the polar axis persists. In attempting continuously to fulfil this desire it acquires a slight upward tilt, which is sufficient to bring the pendulum weight into action. With the weight thus slightly deflected towards the north a moment is applied to the wheel which tends to turn the wheel about the horizontal axis E F in such a way as to bring the end B down again to the horizontal plane. Such a moment, as we know from the fundamental rule of the gyroscope, will actually produce precession about the vertical axis H J, the direction of this precession being such as to cause the end B to move away from the north towards the west.