628 report — 1884. 



Eliminating 6 between these equations we find 



f^" 9 tWK 8 )gf=H (4) 



V ffWa ) dP ' 



which proves that the action of H in generating azitnuthal motion is the same as 

 it would be if a single rigid body of moment of inertia given by the formula (I), 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 sufficiently sheltered enclosure 

 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. Sup- 

 pose for a moment the locality of the experiment to be either the North or South 

 pole, the operation to be performed to bring the gyrostat to rest will not be dis- 

 coverably 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 relatively to the earth, 6 will have a very small constant 

 value ; so small that the inclination of a to the vertical will be quite imperceptible, 

 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 flywheel is 

 just sufficient to cause the axis of the flywheel to turn rouud 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 posi- 

 tion the axis of the gyrostat will be in the North and South line, and, the equili- 

 brium being stable, the direction of rotation of the flywheel must be the same as 

 that of the component 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 zero tor- 

 sional rigidity, the rotational influence would produce, in respect to the equilibrium 

 of the gyrostat with its axis reversed from the position of gyrostatic 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 equilibrium 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 process 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 un- 

 twisted ; it will be found that the torque required to balance it in any oblique 

 position will be proportional to sin <f>. 



The chief difficulty in realising this description results from the great augmen- 

 tation of virtual moment of inertia, represented by the formula (1) above. The 

 paper at present communicated to the section contains calculations on this subject, 

 which throw light on many of the practical difficulties hitherto felt in any method 

 of carrying out gyrostatic investigation of the earth's rotation, and which have led 

 the author to fall back upon the method described by him at Southport, of which 

 the essential characteristic is to constrain the frame of the gyrostat in such a 

 mannfeir as to leave it just one degree of freedom to move. The paper concludes 

 with the description of a simplified manner of realising this condition for a 

 gyrostatic compass: — that is to say, a gyrostat free to move in a plane either 

 rigorously or very approximately horizontal. 



