From these considerations it is argued that by suitably selecting the period of vibration the ballistic deflection can be made for one particular latitude just equal to the difference between the north steaming errors for any initial and any new speed on any course. In practice the latitude selected is 40 deg. north or south. It is found by calculation that the period of vibration which must be given to the compass to secure this dead-beat ballistic deflection is such that in this latitude the compass should oscillate in the same period as would a simple pendulum, the length of which was equal to the radius of the earth. It is for this reason that all modern gyro-compasses have a period of vibration of approximately 85 minutes.[5]
In latitudes other than 40 deg. north or south the ballistic deflection will not be dead-beat, but by taking this latitude as the mean the subsequent oscillation after a change of speed is—in mercantile vessels at least—sufficiently small to introduce no error of great importance, except for purposes of very accurate observation, in which case the observation, if possible, would not be made until the vessel had been running for two or three hours after the last considerable change of speed.
It is to be noticed that the ballistic error may arise when the ship is altering course without altering its speed. Thus, if it is sailing north and makes a sharp turn eastwards or westwards, the speed northwards during the turn falls progressively from a maximum at the beginning to zero at the end of the turn, even although the actual linear speed of the ship remains constant throughout. Turning in either direction is thus equivalent to a deceleration of the ship’s northerly speed, so that an easterly ballistic deflection may be expected in both cases. On the other hand, a turn towards the north from a due east or west course is equivalent to an acceleration of the ship’s northerly speed, and, as a result, a westerly ballistic deflection is produced. In latitude 40 deg. north or south both these ballistic deflections would be dead-beat in a modern gyro-compass, being just sufficient in the one case to eliminate and in the other to apply the appropriate north steaming error.
In a very recent improvement, it is understood, it has been found possible to provide means whereby the ballistic deflection is made dead-beat in all latitudes.
CHAPTER XI
THE QUADRANTAL ERROR
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 E F.
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.