Sept. 25, 1884] 



NA TURE 



525 



towards the north as does the magnetic dipping needle. Thus, 

 if the bearing of the knife-edges be placed east and west, the 

 gyrostat will balance with its axis paa'lel to the earth's axis, 

 and therefore dipping «ith its south en 1 downwards in northern 

 latitudes and its north end downwards in southern latitudes. 

 If displaced from this position and left to itself, it will oscillate 

 according to precisely the same law a-i that by which the mag- 

 netic needle oscillates. If the bearings be turned round in azimuth, 

 the position of equilibrium will follow the same law as does that 

 of a magnetic dipping needle similarly dealt with. Thus, if the 

 line of knife-edges be north and south, the gyrostat will balance 

 with the axis of the fly-wheel vertical, and if displaced from this 

 position will oscillate still according to the same law ; but with 

 directive couple equal to the sine of the latitude into the directive 

 couple experienced when the line of knife-edges is east and west. 

 Thus this piece of apparatus gives us the means of definitely 

 measuring the direction of the earth's rotation, and the angular 

 velocity of the rotation. These experiments will, I believe, be 

 very easily performed, although I have not myself hitherto found 

 time to try them. 



Gyrostatic Model of a Magnetic Com pas:. 



At Southport I showed that a-gyrostat supported frictionlessly 

 on a'fixed vertical axis, with the axis of the fly-wdieel horizontal 

 or nearly so, will act just as does the magnetic compass, tint with 

 reference to "astronomical north" (that is to 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, impeded 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, fine wire, or even by floating it with 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 fly-wheel horizontal, to be hung by a very fine 

 wire attached to its framework at a point, as far as can con- 

 veniently be arranged for, above the centre of gravity of fly-wheel 

 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 fly-wheel 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 supervenes. 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 fly-wheel, 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 fly-wheel and 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 fly-wheel ; 



/', the radius of gyration of the fly-wheel ; 



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

 the centre of gravity of fly-wheel and framework ; 



g, the force of gravity on unit mass ; 



to, the angular velocity of the fly-wheel_; the virtual moment 

 of inertia rjtind a vertical axis is 



WK'(. + ^-) (.) 



V W- K- ag / 



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



tp , the angle between a fixed vertical plane and the vertical 



plane containing the axle of the fly-wheel at any time t ; 

 9, 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 / : 



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



by the bearing wire on the suspended fly-wheel and 



framework. 



By the law of generation of moment of momentum round an 



axis perpendicular to the axis of rotation requisite to turn the axis 



of rotation with an angular velocity dcf>!dt, we have 



lit " 



(2) 



because g~Wa9 is the moment of the couple in the vertical plane 

 through the axis by which the angular motion d<f>/dt in the hori- 

 zontal plane is produced. Again, by the same principle of gene- 

 ration of moment of momentum taken in conneclii .1 with the 

 elementary principle of acceleration of angular velocity, we 

 have 



rf» d ° + WK= pt-,H .... (3) 

 at ai- 



Eliminating 9 between these equations we find 



gWa 



+ WK» )*g = H 



/ dP 



(4) 



which proves that the action of H in generating azimuthal 

 motion is the same as it would be if a single rigid body of 

 moment of inertia given by the formula (1), as said above, were 

 substituted for the gyrostat. Now to realise the gyrostatic 

 model compass : arrange a gyrostat according to the preceding 

 description with a very fine steel bearing wire, not less than 

 5 or 10 metres long (the longer the better ; the loftiest suffi- 

 ciently sheltered inclosure conveniently available should be 

 chosen for the experiment). Proceed precisely as above to 

 bring the gyrostat to rest by aid of the torsion head, attached 

 to a beam of the roof or other convenient support sharing the 

 earth's actual rotation. Suppose for a moment the locality of 

 the experiment to be either the North or South Pole, the opera- 

 tion to be performed to bring the gyrostat to rest will not be 

 discoverably different from what it was, as we first imagined it 

 when the earth was supposed to be not rotating. The only 

 difference will be that, when the gyrostat hangs at rest, rela- 

 tively to the earth, B will have a very small constant value ; so 

 small that the inclination of a to the vertical will be quite im- 

 perceptible, unless a were made so exceedingly small that the 

 arrangement should give the result, to discover which was the 

 object of the gyrostatic model balance described above ; that is 

 to say, to discover the vertical component of the earth's rotation. 

 In reality we have made a as large as we conveniently can ; 

 and its inclination to the vertical will therefore be very small, 

 when the moment of the tension of the wire round a horizontal 

 axis perpendicular to the axis of rotation of the fly-wheel is just 

 sufficient to cause the axis of the fly-wheel to turn round with 

 the earth. Let now the locality be anywhere except at the 

 North or South Pole ; and now, instead of bringing the gyrostat 

 to rest at random in any position, bring it to rest by successive 

 trials in a position in which, judging by the torsion head and 

 the position of the gyrostat, we see that there is no torsion of 

 the wire. In this position the axis of the gyrostat will be in the 

 north and south line, and, the equilibrium being stable, the 

 direction of rotation of the fly-wheel must be the same as that of 

 the compotent rotation of the earth round the north and south 

 horizontal line, unless (which is a case to be avoided in practice) 

 the torsional rigidity of the wire is so great as to convert into 

 stability the instability which with 2110 torsicnal rigidity the 

 rotational influence would produce in respect to the equilibrium 

 of the gyrostat with its axis reversed from the position of gyro- 

 static stability. It may be remarked, however, that even though 

 the torsional rigidity were so great that there were two stable 

 positions with no twist, the position of gyrostatic unstable equi- 

 librium made stable by torsion would not be that arrived at : the 

 position of stable gyrostatic equilibrium, rendered more stable 

 by torsion, would be the position arrived at, by the natural pro- 

 cess of turning the torsion head always in the direction of finding 

 by trial a position of stable equilibrium with the wire untwisted 

 by manipulation of the torsion head. Now by manipulating the 

 torsion head bring the gyrostat into equilibrium with its axis 

 inclined at any angle, (/>, to that position in which the bearing 

 wire is untwisted ; it will be found that the torque required to 

 balance it in any oblique position will be proportional to the sin 

 <p. The chief difficulty in realising this description results from 

 the great augmentation of virtual moment of inertia, repre- 

 sented by the formula ( I ) above. The paper at present commu- 

 nicated to the Section contains calculations on this subject, which 



