2342 
NATURE 
[Marcu 4, 1915 

the companion component about the horizontal there 
is wcosl. Thus at London the component about the 
vertical is 0°78 of w, and the period of rotation about 
the vertical is about 30°77 hours of sidereal time. 
Lord Kelvin’s method of measuring wsinl con- 
sists in supporting a gyrostat on knife-edges attached 
to the projecting edge of the case, so that the gyro- 
stat without spin rests with the axis horizontal or 
nearly so. For this purpose the line of knife-edges is 
laid through the centre of the flywheel at right 
angles to the axis, and the plane of the knife-edges 
is therefore the plane of symmetry of the flywheel 
perpendicular to the axis. The knife-edges are a 
little above the centre of gravity of the instrument, 
which we suppose in or nearly in that plane, so that 
there is a little gravitational stability. The azimuth 
of the axis is a matter of indifference, as any couple 
due to the component of rotation about the hori- 
zontal is balanced by an equal couple furnished by 
the knife-edge bearings. 
At points in a line at right angles to the line of 
Kknife-edges, and passing through it, two scale-pans | 
are attached to the framework, and by weights in 
these the axis of the gyrostat (without spin) is 
adjusted, as nearly as may be, in a horizontal position 
which is marked. The gyrostat is now removed, to | 
have its flywheel spun rapidly, and is then replaced. 
It is found that the weights in the scale-pans have 
to be altered now to bring the gyrostat back to the 
marked position. From the alteration in the weights 
the angular speed about the vertical can be cal- 
culated. 
To fix the ideas, let the gyrostat axis be north 
and south, and let the spin to an observer, looking 
at it from beyond the north end, be in the counter- 
clock, or positive direction. The rotation of the earth 
about the vertical carries the north end of the axis 
round towards the west, and 
momentum is being produced about a horizontal axis 
drawn westward, at a rate equal to Cnwsinl, where 
C is the angular momentum of the flywheel. If the 
sum of the increase of weight on one scale-pan and 
the diminution (if any) in the other be w, and a be 
the horizontal distance between the points of attach- 
ment of the scale-pans, we have 
Cnowsinl=w ga. 
Thus if C and n are known, wsinl, or w, can be 
calculated. 
Lord Kelvin does not give any figures as to the 
forces to be measured in a practical experiment; but 
I can supply these. We may take the mass of a 
small flywheel as 400 grammes, its radius of gyra- 
tion as 4 cm., and its speed of revolution, if high, as 
200 revolutions per second. If we take a as ro cm. 
we obtain for London the equation 
2m x 078 
80160 
This gives w=o'047 gramme, or 47 milligrammes. 
It would require careful arrangements to carry out 
the experiment accurately, but the idea is clearly not 
unpractical. With some of the new gyrostats that 
we now have, the mass of the wheel is as much as 
2000 grammes, and the radius of gyration is 
about 75 cm. These numbers bring w up to 0-82 
gramme, at the same speed. 
If the gravitational stability of this gvrostatic 
balance be removed, that is, the line of knife- 
edges be made to pass accurately through the centre 
of gravity of the system of wheel and framework, 
and the axis of the wheel be placed in a truly north 
and south vertical plane, so that the knife-edges are 
horizontally east and west, the gyrostat will be in 
NO. 2366, VOL. 95| 
400 X 4? x 4007 x =10X 981 x w. 
| 

therefore angular | 
stable equilibrium when the axis is parallel to the 
earth’s axis, and is turned so that the direction of 
rotation agrees with the rotation of the earth. For 
we have then simply the experiment, described above, 
of the gyrostat mounted on trunnions resting on 
bearings attached to a tray which is carried round 
by the experimenter. The axis of the gyrostat was 
at right angles to the tray, and we saw that when 
the tray, held horizontally, was carried round in 
azimuth the equilibrium of the gyrostat was stable 
or unstable, according as the two turnings agreed 
or disagreed in direction. In the present case the 
tray is the earth, the position of the axis of rotation 
parallel to the earth’s axis replaces the vertical posi- 
tion, and the earth’s turning the azimuthal motion. 
If displaced from the stable position the gyrostat will 
oscillate about it in the period 27¥A/Cnw, where A 
is the moment of inertia about the knife-edges, and 
the other quantities have the meanings already 
assigned to them. 
If the line of knife-edges be north and south, the 
vertical will be the stable, or unstable, direction of 
the axis of rotation, and there will be oscillation 
| about the stable position in the period 
2nVA/Cnosinl. 
The gyrostat thus imitates exactly the behaviour 
of a dipping needle in the earth’s magnetic field, 
and thus we have Lord Kelvin’s gyrostatic model of 
the dipping needle. 
It is right to point out that these arrangements 
| were anticipated by Gilbert’s barogyroscope,’? which 
rests on precisely the same idea, and applies it in a 
similar manner. 
IX.—Gyrostatic Compass. 
At Montreal Lord Kelvin described a’ ‘‘ gyrostatic 
model of a magnetic compass.’? This was one of 
| his gyrostats hung, with its axis of rotation hori- 
zontal, by a long fine wire, attached to the frame- 
work at a point over the centre of gravity of the 
system, and held at the upper end by a torsion-head 
capable of being turned round the axis of the wire. 
By means of this torsion-head any swinging of the 
gyrostat in azimuth round the wire was to be checked 
until, when the head was left untouched, the gyro- 
stat hung at rest. 
In small azimuthal oscillations of the gyrostat 
about the axis of the wire, the wire being fixed to 
the gyrostat at the lower end and held by the head 
at the upper, the virtual moment of inertia of the 
| gyrostat about the wire is greatly enhanced by the 
rotation of the flywheel If there were no rotation 
the moment of inertia would be A; with rotation it 
is virtually A(1+C?n?/A Mag), where M is the 
whole suspended mass, a the distance of the point 
of attachment of the wire above the centre of 
gravity of the mass M. This will be found 
proved very simply in the Mathematical Appendix 
[see Journal, 1.E.E.] to this lecture. It will 
| be shown, moreover, that when the whole motion 
is considered—the tilting motion as well as the 
azimuthal—it appears that there are two funda- 

mental periods of vibration. There is the long period 
due to the slight torsional rigidity of the long wire, 
and the enhanced moment of inertia pointed out by 
Lord Kelvin, and also a short period, the shortness 
of which is most properly to be reckoned as due to 
virtual diminution of the moment of inertia of the 
gyrostat, in the tilting motion, in exactly the same 
ratio as the other moment of inertia is increased. 
Both these periods are separately possible, and in 
12 “ Mémoires sur divers problémes,” etc. Annales de la Société Scien- 
tifigue, Bruxelles, 1877—8. 
