Let a stone be tied to the end of a string and be flung forward while the free end of the string is held in the hand. Let the hand holding the free end be moved forward at first with the same speed as the stone in its flight, and then let the hand be drawn back. We know that as soon as the string becomes taut—if it were not already fully extended—the hand would feel a pull exerted on it in the direction in which the stone was initially projected. This pull arises from the fact that the stone has been given momentum in the direction of its flight, and while this momentum is being taken out of it by the backward pull of the hand and new momentum in the backward direction is being communicated to it the stone reacts forcibly on the string and therefore on the hand.

In the gyro-compass the pendulum weight may be likened to the stone, the stirrup carrying the weight to the string, and the spinning wheel to the hand. While the ship is steaming at a uniform speed, the system is in the condition existing just after the stone has been projected from the hand, and while the hand is following it with equal speed. A change in the ship’s speed—a reduction of its speed to be quite correct in our analogy—is comparable with the drawing back of the hand. The pull of the stone exerted on the hand at this instant is represented by the tendency of the weight to continue moving at the former speed, and the resultant “kick” which by its attempt to do so it applies to the spinning wheel. This kick is communicated to the spinning wheel through stiff members and rigid connections, and not through a flexible string. It really acts at the centre of gravity of the pendulum weight, and is therefore felt by the spinning wheel, not as a straight pull, but as a force tending to turn the wheel about the horizontal axis EF.

The kick of the weight when the ship changes speed on any course having a north or south component produces, as we have seen, a transient ballistic error which may influence the correctness of the compass readings for some hours after the change of speed has been completed. We have now to show that similar kicks occur under other conditions, and may similarly affect the accuracy of the readings.

The subject which we are about to discuss is the so-called quadrantal error which, unless steps are taken to eliminate it, appears in the compass readings when the ship is sailing in a rough sea on any course other than due north, south, east, or west. It is caused by the rolling and pitching of the ship. The efforts made to eliminate it have influenced the evolution of the gyro-compass to a greater extent than probably those directed towards overcoming or allowing for all the other errors combined to which the device is open.

The gyro-compass on board ship is usually, although in no way necessarily, mounted above the vessel’s metacentres—transverse and longitudinal—about which rolling and pitching take place. In Fig.29 we show a section, looking aft, of a vessel steaming due north with a gyro-compass mounted on its deck. It is clear that, so far as the weight S is concerned, it may be regarded when the ship rolls as the bob of an inverted pendulum vibrating in an east and west plane, through an angle equal to the angle of the ship’s roll. From what we have already said, it will be seen that at the end of a roll to port the hand holding the string attached to the flying stone, hitherto following it, begins its withdrawal to the east. The weight checked in its movement to the west communicates a kick to the sensitive element. This kick, a westward force, is applied at the centre of gravity of the weight, and clearly does nothing more than throw a stress on to the journals at EF and HJ, and through the square frame on to the deck. It does not therefore affect the direction in which the gyro-axle is pointing. If, as is actually the case in practice, the equivalent of the square frame is not fixed directly to the deck, but is mounted inside the binnacle on athwartship and longitudinal gimbals, the square frame will swing on the longitudinal gimbal axis. The weight S will, however, still act during the ship’s roll as an inverted pendulum, as at ZZ, in which the bob remains more or less parallel with its original direction. Under these conditions the kick of the weight at the end of a roll to port will rotate the frame in the direction of the curved arrow about the longitudinal gimbal axis. As this axis is coincident or parallel with the gyro-axle, the gyro-axle is not moved otherwise than parallel with itself either by the actual roll of the ship or by the kick of the weight at either out position. Thus a ship steaming north may roll with impunity without affecting the accuracy of the direction in which the gyro-axle is pointing.

It is clear that rolling on a due south course or pitching on a due east or west course is similarly without effect on the gyro-compass.

Now consider the case of the ship rolling when on a due west course (Fig.30). It is clear that here again the weight S may by itself be considered as the bob of an inverted pendulum vibrating through the angle of the ship’s roll, this time in a north and south plane. The weight S has, in addition, its own pendulum action on the horizontal axis EF, and tends to keep the gyro-axle horizontal throughout the roll.

At the end of a roll to port the kick of the weight S, a southerly force acting at the centre of gravity of the weight, tends to turn the spinning wheel in the direction R about the horizontal east and west axis EF. This kick is clearly equivalent to the application of an upward force at the end C of the axle or a downward force at the end B, and therefore, as we know, will cause the axle not to turn about EF, but to precess the north end B towards the west about the axis HJ.

The roll to starboard now takes place, and at its end the kick of the weight occurs in the northerly direction. This kick will tend to turn the spinning wheel about the east and west axis EF in the direction T, and is clearly equivalent to the application of an upward force at the end B of the axle. Such a force, as we know, will cause the end B of the axle to precess towards the east.

It will thus be seen that the western deflection of the axle produced by the precession which occurs at the end of the roll to port is counterbalanced and automatically eliminated by the eastern deflection produced by the precession occurring at the end of the roll to starboard. The only effect on the compass caused by the rolling of the ship is therefore a vibration of the axle about the north and south direction. We should really say that the only effect is the application to the sensitive element of a vibratory influence in tune with the rolling of the ship. As the ship’s period is very small compared with that of the compass about the axis HJ—about 5 to 12 seconds as compared with about 85 minutes—this vibratory influence practically fails to disturb the steadiness with which the axle points to the north.

It is clear that pitching of the ship when on a due west course has the same effect—or lack of effect—on the compass as rolling on a due north course, and that pitching on a due north course has the same effect as rolling on a due west course. It is further clear that due south and due east courses are similar in this respect to due north and due west courses. Thus when the ship is on a cardinal course neither pitching nor rolling disturbs the axle of the spinning wheel from its north resting position.

Matters are quite otherwise, however, when the course is an inter-cardinal, or quadrantal, one. The discovery of this unlikely fact was in large part due to the investigations of the Compass Department of the British Admiralty. It does not appear to have been known, or at least its full significance does not seem to have been appreciated, in the earlier days of gyro-compass construction. At least one early design of compass—the AnschÜtz of 1910—did not include means of eliminating or allowing for the “quadrantal error,” and soon became obsolete as a result primarily of the omission. That the later design of AnschÜtz compass successfully overcomes the difficulties introduced by this error is shown by the admitted excellence of the navigation of the German submarines, which vessels were universally fitted with compasses of this design, and which, like all their class, suffer much from rolling and pitching.

So far we have been able to use a very simple model to demonstrate the properties and errors of the gyro-compass. We have now reached a subject which, if it is to be explained correctly, requires us to adopt a model of a more elaborate nature, one in which the outer square frame of our simple model is itself mounted on a pair of gimbal axes. We have already referred to this type of mounting in our second chapter and above in connection with Fig.29. In Fig.31 we show it with the pendulous weight added to the sensitive element. We may take it that the axis TU, about which the entire compass system may swing, is parallel with the longitudinal centre line of the ship, and that the frame Y is fixed to the deck.

In order to make quite clear the difference between the two forms of mounting, we give in Figs. 32 and 33 two corresponding sets of views showing the compass system on a ship while steaming due north and rolling. With the simple mounting (Fig.32) the weight S constantly remains radially below the centre of the spinning wheel, and the kicks which it applies at the out port and out starboard positions merely stress the compass mountings. With the more elaborate mounting (Fig.33) the weight tends to remain constantly in the vertical below the centre of the spinning wheel, but the kicks occurring at the out positions, if the rolling is continued for any length of time, cause it to swing beyond the vertical, so that as the ship rolls the whole compass system acquires an oscillation on the axis TU (Fig.31) in tune with the rolls. It is to be noted that the period of vibration of the compass system about the axis TU is very much less than that of the sensitive element about the axis HJ. The latter, as we have seen, is about 85 minutes, and is determined, in part at least, by the high speed of the spinning wheel. About TU the vibration, however, does not call any gyroscopic force into play, for it takes place without causing the axle of the spinning wheel to alter its direction. The period of this vibration is therefore not affected by the speed of the spinning wheel. It may be quite small—from one to two seconds—and therefore comparable with the ship’s rolling period. If, then, no steps are taken to prevent it, the compass system as a whole will acquire a swing on the axis TU (Fig.31) when the ship is steaming north—or south—and rolling. Its period of vibration on this axis is not strictly that which it would have were it set oscillating about the axis TU with this axis mounted on a fixed frame, for the vibration is not a free one, but is a forced vibration under the impulses communicated by the rolling of the ship. Whatever may be the exact free period of swing of the compass system on the axis TU, so long as it is somewhere near that of the ship’s roll or a small sub-multiple of the rolling period, the tendency is for the compass swings to settle down in such a way that, as represented in Fig.33, the pendulum weight S reaches its extreme out positions simultaneously with the ship’s arrival at its extreme port and starboard heels. In the course of one complete roll of the ship, however, the compass system may make more than one complete vibration about TU.

It is clear that whether the bob S remains radially below the centre of the wheel, as in Fig.32, or remains vertically below it, or swings past the vertical, as in Fig.33, the effect of rolling when the ship is on a due north course is as established in connection with Fig.29. Either a tendency to rotate the wheel about an axis coincident with its axle or an actual rotation of the wheel about this axis results from the kicks of the weight S at the out positions. Whichever it is, there is no gyroscopic effect called into play, and the axle is not subjected to any influence causing it to move away from the due north direction.

We have now to discuss what happens if the vessel rolls or pitches when sailing on an inter-cardinal or quadrantal course.

In Fig.34 we show in plan a vessel steaming on a north-west course. When the ship rolls the compass oscillates in the path AD, a path, that is to say, at right angles to the ship’s centre line and curved upwardly by reason of the fact that the compass is mounted above the rolling centre of the ship. The axle of the compass during this movement tends to point steadily towards the north under the action of the directive force. The wheel therefore oscillates from A to D with its axle askew relatively to the path of oscillation, and not at right angles to it, as in Fig.29, or parallel with it, as in Fig.30. The oscillation AD can, however, be resolved into two components, namely, a north and south oscillation DK, in which the axle is parallel with the path, and an east and west oscillation KA, in which it is at right angles to the path, both paths being curved upwardly.

The oscillation DK, taken by itself, is clearly exactly equivalent to that shown in Fig.30. The southward kick of the weight at the “out position” D (Fig.34) just neutralises the disturbing effect of the northward kick at the “out position” K. Similarly, the oscillation KA, taken by itself, is exactly equivalent to that shown in Fig.29 or its modification, Fig.33. The eastward kick of the weight at A and the westward kick at K cannot do more than turn the mounting round the axle of the spinning wheel. They do not tend to change the direction of the axle.

Hence, taken separately, each component oscillation DK and KA is without disturbing effect upon the direction in which the axle is pointing. Taken together, however, their united effect is not the sum of their separate effects, as we might hastily assume. We must not suppose that the north and south kicks proper to the north and south component of the oscillation are without effect upon the east and west component or vice versa. At the out position D of the component oscillation KD the weight is not only kicked southwards, but also receives a westward kick from the other component oscillation AK. At K it receives both a northward kick and an eastward kick. Similarly, at the out position K of the component oscillation AK the weight is kicked not only to the west but also to the south, by virtue of the oscillation from K to D. At the other out position A of this component the weight is kicked eastwardly, and also to the north. If we take cognisance of these additional kicks, we may treat the component oscillations separately and add the separate results to get the true effect of the actual oscillation AD.

Returning, then, to Fig.30, let us suppose that in the oscillation there shown, the equivalent of the component KD, we apply a westward kick to the weight S when the ship reaches its port out position—that is to say, a kick out of the paper—and that at the starboard out position it receives an eastward kick into the plane of the paper. It is clear that these additional kicks do nothing more than tend to rotate the wheel about its axle or an axis coincident or parallel with its axle. They cannot therefore call any gyroscopic effect into action, and as a result do not disturb the direction in which the axle is pointing. The component oscillation DK, even with the extra kicks added, is thus harmless.

Taking Fig.33 rather than Fig.29 as the equivalent of the component oscillation AK, let us suppose that at the port out position the weight is subjected to an additional kick to the south—that is, into the plane of the paper—and that at the starboard out position it receives an extra kick to the north, or out of the plane of the paper. The southward kick on the weight at the port out position will tend to cause the end B of the axle to turn down towards J, but, in accordance with the fundamental gyroscopic rule, the actual motion of the sensitive element will not be a rotation about the axis EF, but a precession about the axis HJ, the end B of the axle moving in the direction K. Similarly, the northward kick at the starboard out position will tend to cause the end B of the axle to rise towards H, and will therefore produce precession in the direction L. Now these two precessional movements, K and L, can be resolved into horizontal and vertical components NM and QP respectively, as shown separately, and of these components it is clear that the movement N cancels the movement Q. On the other hand, the two components M and P are in the same direction, and therefore do not cancel each other. Their effect is cumulative, and with each succeeding roll the end B of the axle tends to rise higher and higher in the true vertical plane BR. It may be explained, perhaps, that this motion is possible, in spite of the fact that there is no actual axis in the mounting of the compass at right angles to the true vertical plane, except at the instant when the compass is passing through the even keel position in the course of its swing from side to side. As the rolling of the ship starting from zero increases up to a more or less steady value, the end B of the axle rises slowly at first and then at a constant rate, as exaggerated at T. Such of this upward movement as occurs while the compass is passing through the even keel position can be effected by means of a rotary motion purely about the axis EF. Elsewhere in the swing of the compass from side to side it is accommodated by a partial rotation on EF and a partial rotation about HJ. These two axes between them permit the end B of the axle to rise in a vertical plane at any point of the compass swing just as effectively as would an axis which was at right angles to the vertical plane BR.

To appreciate the significance of this vertical rise of the north end B of the axle, let us consider the condition of the compass as it is passing through the even keel position. The elevation of the end B out of the horizontal plane causes the weight S to swing forward towards the north, and therefore to apply a turning moment to the spinning wheel about the axis EF. This turning moment will tend to bring the end B down again to the horizontal, but, as we know, the actual motion produced will be a precession about the axis HJ, the end B moving westwards. Looking at Fig.34, it will thus be seen that the net effect of the rolling of the ship is to deviate the axle of the compass in the direction shown at V—that is to say, in the direction required to set the plane of the spinning wheel parallel with the plane of the rolling by the shortest possible course—in our example by a rotation westward through 45 deg. With the axle so deflected the earth’s rotation, of course, calls a directive force into play, tending to restore the axle to the north and south line, but if the ship is rolling violently the deviating force will be much stronger than the directive force until a very considerable angle of deflection is reached. When the balance is struck, the axle settles down with a steady deviation towards the west, which will remain constant as long as the rolling is maintained. In practice the violence of the rolling varies almost from roll to roll, for it represents a conflict between the natural period of rolling of the ship and the period of the waves. As a consequence, the deviation of the compass varies somewhat during the rolling, but it is always westwardly if the course is in the north-west quadrant.

A little consideration will show that if the ship is steaming towards the north-east the additional kick on the weight S (Fig.33) at the port out position will be towards the north and at the starboard out position towards the south. The end B of the axle is therefore precessed vertically downwards instead of upwards, and as a result the deviation of the axle is eastwardly. In general, if the ship is steaming on any course in the north-west or south-east quadrant, the deviation caused by her rolling will be towards the west. It will be towards the east if the course is in the north-east or south-west quadrant. The effect of pitching on a quadrantal course is to cause a deviation of the axle in the direction opposed to that of the deviation caused by rolling, so that if the vessel be both rolling and pitching, the deviation is somewhat less than it would be if rolling only had to be considered.

It may perhaps be thought that the quadrantal error is an effect produced by the double gimbal mounting of the compass, and that had we adhered to our simple model, as shown in Fig.32, it would not arise. This is not so. If Fig.32 be taken as representing, after the manner of Fig.33, the north view of the compass on a ship steaming due north-west, it will be seen that the southern kick at the port out position causes the axle to precess in the direction K and the northern kick at the starboard out position in the direction L. Resolving these movements as before, we see that, while the horizontal components NQ again cancel, the vertical components MP are, as in Fig.33, cumulative in effect, but this time they result in the end B of the axle moving downwards, and therefore finally cause it to precess eastwards instead of westwards, as before. The quadrantal error is therefore not eliminated, but merely reversed in direction by adopting the simpler mounting for the compass.