326 VON ARX [CHAP. 16 



The Schuler-tuning condition is satisfied for a given station when T =2n\/(Rlg), 

 where R is the local radius of the earth. 



Schuler tuning of a reference vertical (Wrigley, 1941, 1950) becomes of 

 interest when the motion of a ship generates geocentric angles which approxi- 

 mate the order of accuracy required of the vertical reference. Since one minute 

 of arc is related by definition to the length of the nautical mile, it is readily 

 apparent that a second of arc approximates the length of a ship. Even hove-to, 

 a ship may easily move a distance comparable with its own length in the few 

 minutes needed to make an observation. 



A Schuler-tuned gyropendulum erects to vertical through a succession of 

 84-min oscillations. When these oscillations are larger in amplitude than the 

 desired accuracy of vertical indication, damping must be applied. -But since, 

 in existing systems, the damping function does not satisfy the Schuler-tuning 

 condition (the dash-pot of the equivalent simple pendulum is not at the center 

 of the earth) such a system responds to changes in the motion of the supporting 

 vehicle. For this reason it is probably desirable that the system be undamped 

 when measurements are in progress and to average observations through the 

 period of at least one conical oscillation. 



A further qualification of Schuler-tuned systems is that they tend to com- 

 pensate for geographic motion of the supporting vehicle in a geocentric mode. 

 To provide a continuous indication of the local-gravity vertical, it is necessary 

 to re-tune the system to correspond to the local radius of the earth according 

 to some idealized model of the earth's figure, and, for this reason, to know the 

 geographic position of the pendulum. For the oceanographic purposes to be 

 discussed, it seems better to employ a system that yields gravity vertical more 

 directly. 



To this end it has been shown that if a physical system has a period of conical 

 oscillation which is very great compared with the period of roll and pitch of a 

 ship, but less than 84 min, it will still tend to average through horizontal 

 disturbances and gradually, with isotropic damping, tend to seek the gravity 

 vertical at sea. Motion of the pivot point will misalign the spin axis of the 

 gyropendulum with the direction of local gravity, thereby producing a gravita- 

 tional torque on the pendulum which will cause it to precess. If the translation 

 is steady in inertial space and angular equilibrium has been attained, it is 

 possible for a gyropendulum to maintain a fixed attitude relative to both 

 gravity and the path of motion of its pivot similar to that shown in the right or 

 left half of Fig. 1. This behavior of a gyropendulum causes its length to sweep 

 out a flat cone having an axis normal to the plane of its orbit. Schuler (1923) 

 has pointed out that two gyropendula spinning in opposite directions, but 

 having otherwise identical properties, could be used to generate two cones 

 sharing the same base but with their apices oppositely directed along the 

 central normal to their circular orbit, as shown in Fig. 1. Were the angle 

 between the two translating gyropendula to be continuously bisected, . local 

 gravity vertical would be indicated so long as the rate of translation was slow 

 compared with the tangential velocity of the earth at the latitude of the system. 



