72 Mr Bennett, The Rotation of the Non-Spinning Gyrostat 



§ 4. For the special case in which d is constant, so that the 

 axis of the gyrostat describes a circular cone, the rotation is stated 

 by Sir George Greenhill* to be 27r — (conical angle described by 

 the axle), as against the solid angle itself found above. The differ- 

 ence of sign of the latter can be accounted for by a reverse sign- 

 convention: but the term 27t is unnecessary if 27r is implied as a 

 modulus, and it appears to be wrong if the precise angle of turning 

 is intended. If, specially, the axis of the gyrostat described only 

 a small cone, then the angle of consequent rotation is certainly a 

 small angle, and not an angle nearly equal to four right angles. 



He adds the remark that the movement "can be shown ex- 

 perimentally with a penholder held between the fingers and moved 

 round in a cone by the tip of a finger appHed at the end." But the 

 illustration is inapt; for the creep of the penholder occurs in the 

 sense opposite to that of the conical movement. The body-axode 

 is a circular cone and not a plane, and it rolls inside a shghtly 

 larger circular cone as space-axode; and hence the reverse move- 

 ment. 



§ 5. The movement of the non-spinning gyroscope here con- 

 sidered is not yet among those that are familiarly recognised, though 

 it has important practical applications and deserves to rank as a 

 dynamical commonplace. Bodies suspended from a point on an 

 axis of symmetry behave in the same way and for the same reason. 

 No matter how the point of suspension may be moved about, and 

 no matter what complicated conical movement is consequently 

 executed by the axis, the applied forces have no moment about the 

 axis, and the spin remains zero if originally zero. The resultant 

 rotation is then given, as above, by the solid angle of the cone 

 described by the axis. 



Aeroplane compasses, in particular, are found to keep their 

 cards practically parallel to the floor, under the combined action 

 of gravity and lateral acceleration, during a banked turn of the 

 aeroplane. Hence, from inertia alone, and apart from all other 

 sources of control or disturbance, the compass-card would be 

 rotated, as a consequence of the turn, through an angle equal to 

 the solid angle described by the normal to the card. For an angle 

 of banking a and a change of course ^ the solid angle is not much 

 less than (1 — cos a)^ if the banking is taken and left quickly; and 

 for very steep banking this angle is nearly equal to the change of 

 course itself, and the card would almost appear to "stick." As 

 compared with considerations of magnetic disturbance due to the 

 vertical component of the earth's field, and of mechanical disturb- 

 ance due to rotation of the bowl and liquid, the pure inertia efiect 



* Advisory Committee for Aeronautics. Reports and Memoranda, No. 146, 

 Report on Gyroscopic Theory, p. 13, § 14. 



