190   NAVIGATIONAL COMPASSES

At the equator R = 20,925,000 ft., and g = 32.086 ft. per sec. per sec. Hence, at the equator, To[= 5068 sec.] = 84.4 min. Since at other latitudes the radius of curvature of the earth is slightly less and g is slightly greater than at the equator, the period of the compass at places either north or south of the equator should be somewhat less than at the equator.

Since (125) must hold when the period of azimuthal vibration of the undamped compass is such that there is zero ballistic deflection error, we see that, at any latitude d, the period can be adjusted to the desired value by varying either hs or mx. Hence, a gyro-compass at latitude X will have zero ballistic error when either hs or mx is adjusted so that the ratio

The mx in this equation is called the " pendulous factor."

The period of every gyro-compass is made of such a value that when the instrument is at a selected latitude, the ballistic error is zero. The desired period is obtained by having proper values of the angular momentum of the gyro and of the pendulousness of the sensitive element or the mass of active mercury in the ballistic.

114. The Ballistic Damping Error. - While the meridian component of a ship's velocity is being accelerated, either by a change of speed or of course, there will be a tilting of the gyro-compass except when the acceleration is perpendicular to the spin-axle. This is due to forces impressed by the damping mechanism and to the precession in azimuth produced by the pendulousness. When the degree of pendulousness is correct for a particular latitude, then the gyro-compass will give zero ballistic deflection error at that latitude during the time the ship's velocity is accelerating. At any other latitude, the gyro-compass will show a ballistic deflection error during the time the velocity of the ship is accelerating. The tilt of the spin-axle associated with the acceleration of the ship's velocity is superposed on the tilt due to the rotation of the earth. The torque producing this additional tilt ceases when the acceleration ceases but the spin-axle starts oscillating back and forth through the equilibrium position of the spin-axle for the particular velocity and latitude of the ship.

The oscillation of the gyro-compass may continue for an hour or more after the acceleration of the ship's velocity has ceased. The device employed to damp the oscillation exerts a torque on the

NATURAL ERRORS   I

spin-axle which displaces the settling point from the normal equilibrium position. The maximum deflection of the spin-axle from the equilibrium position due to this cause is called the maximum ballistic damping error. In the case of any gyro-compass, the damping of the vibration of the gyro-axle produces an error after the ship has completed a change in course or speed. This ballistic damping error is most marked after the ship, steaming at full speed, has completed a turn of 90 degrees or more. It attains a maximum value in about 20 minutes after the velocity of the ship has become constant. The magnitude of the damping error increases with increase in the acceleration of the ship. Much greater accelerations are produced by sudden turns than by any possible change in the speed of the ship. Since this error is small while the ship is on a straight course but may be large when the ship is making a turn, it is also called the ballistic turning error. It is also called the damping acceleration error.

The ballistic damping error and the accompanying oscillation of the spin-axle can be prevented by stopping the operation of the damping device during the turning of the ship.

115. The Compass Error Due to Rolling of a Ship When on an Intercardinal Course. The Quadrantal or Rolling Error. - A pendulous gyro-compass on a ship that is rolling or pitching is acted upon by a force that has a maximum value at the end of a roll or pitch, and another force that has a maximum value when the pendulous system is passing through its equilibrium position. The first of these forces will be considered in the present Article. The second will be considered in the following Article.

A ship's compass is supported in a Cardan gimbal mounting consisting of two horizontal rings, one capable of rotation about an axis parallel to the keel of the ship and the other about a transverse axis. In so far as freedom to turn in any direction is concerned, a gyro-compass on board ship is equivalent to a gyro-wheel mounted in five rings. For simplicity of representation, in the present Article, the pendulous gyro-compass will be represented by a gyro-wheel in a casing free to turn about any axis through its center and carrying an additional mass attached to the lower side of the casing. This mass that causes the gyro and casing to be pendulous we shall call " the pendulous mass."

Consider the effect on the direction of a gyro-compass produced by rolling or pitching of a ship about an axis that is perpendicular to the spin-axle. In Fig. 147, the keel of the ship is east and