TRANSACTIONS OF SECTION A. 627 



say, rotational North) instead of ' magnetic North.' I also showed a method of 

 mounting a gyrostat so as to leave it free to turn round a truly vertical axis, im- 

 peded by so little of frictional influence as not to prevent the realisation of the 

 idea. The method, however, promised to be somewhat troublesome, and I have 

 since found that the object of producing a gyrostatic model of the magnetic 

 compass may, with a very remarkable dynamical modification, be much more 

 simply attained by merely suspending the gyrostat by a very long line wire or even 

 by floating it wath sufficient stability on a properly planned floater. To investigate 

 the theory of this arrangement let us first suppose a gyrostat with the axis of its 

 flywheel horizontal, to be hung by a very fine wire attached to its framework at 

 a point, as far as can conveniently be arranged for, above the centre of gravity of 

 flywheel and framework, and let the upper end of the wire be attached to a 

 torsion head, capable of being turned round a fixed vertical axis as in a Coulomb's 

 torsion balance. First, for simplicity, let us suppose the earth to be not rotating. 

 The flywheel being set into rapid rotation, let the gyrostat be hung by the wire, 

 and after being steadied as carefully as possible by hand, let it be left to itself. If 

 it be observed to commence turning azimuthally in either direction, check this 

 motion by the torsion head; that is to say, turn the torsion head gently in a 

 direction opposite to the observed azimuthal motion until this motion ceases. Then 

 do nothing to the torsion head, and observe if a reverse azimuthal motion super- 

 venes. If it does, check this motion also by opposing it by torsion, but more gently 

 than before. Go on until when the torsion head is left untouched the gyrostat 

 remains at rest. The process gone through will have been undistinguishable from 

 what would have had to be performed if, instead of the gyrostat with its rotating 

 flywheel, a rigid body of the same weight, but with much greater moment of 

 inertia about the vertical axis, had been in its place. The formula for the 

 augmented moment of inertia is as follows. Denote by — 



W, the whole suspended weight of flywheel aud framework, 



K, the radius of gyration round the vertical through the centre of gravity of 

 the whole mass regarded for a moment as one rigid body, 



w, the mass of the flywheel, 



7c, the radius of gyration of the flywheel, 



a, the distance of the point of attachment of the wire above the centre of 

 gravity of flywheel and framework, 



g, the force of gravity on unit mass, 



o), the angular velocity of the flywheel ; the virtual moment of inertia round 

 a vertical axis is 



WK ; (l + ^) (1) 



The proof is very easy. Here it is. Denote by — 



<£, the angle between a fixed vertical plane and the vertical plane containing 



the axis of the flywheel at any time t, 

 8, the angle (supposed to be infinitely small and in the plane of (p) at which 



the line a is inclined to the vertical at time t, 

 H, the moment of the torque round the vertical axis exerted by the bearing 

 wire on the suspended flywheel and framework. 

 By the law of generation of moment of momentum round an axis perpen- 

 dicular to the axis of rotation recpiisite to turn the axis of rotation with an angular 

 velocity ctyjdt, we have 



ic?c-<o & = f/^Vad (2) 



because gVfad is the moment of the couple in the vertical plane through the axis 

 by which the angular motion dcfrjdt in the horizontal plane is produced. Again, by 

 the same principle of generation of moment of momentum taken in connection 

 with the elementary principle of acceleration of angular velocity, we have 



wlc-co d f : + M ! f$ = H (3) 



dt dt' , : 



