20 
NATURE 
[Marcu 4, 1915 ° 

we have a solution giving oscillations about the 
vertical, in the period 27/A/oN: the equilibrium 
is stable. When, however, o is reversed the product 
must be given the opposite sign, and we get a solution 
in real exponentials, starting falling-away from the 
upright position, which is continued until the oppo- 
site (stable) position is attained. N has now also been 
reversed in space, and the product wN in the differ- 
ential equation is again positive. 
As I have already stated, the time integral of the 
turning motive about the vertical required by the 
gyrostat from the frame constraining it to move round 
in azimuth is 2N; that is, 2Cm, where C is the 
moment of inertia of the flywheel about its axis. 
There is thus at each instant of the turning in azi- 
muth before the inversion has been completed a couple 
required from the frame, and this couple is greater 
the greater the angular speed n of spin. 
The couple arises thus. Let the gyrostat axis have 
been displaced from the vertical through an angle 
6 about the trunnion axis. In consequence of the 
azimuthal motion, at rate w, say, the outer extremity 
of the axis of angular momentum is being moved 
parallel to the instantaneous position of the line of 
trunnions, and thus there is rate of production R 
of angular momentum about that line; but there 
being no applied couple about the trunnions, the gyro- 
stat must begin to turn about the trunnions to 
neutralise R. This turning tends to erect or to 
capsize the gyrostat according as the spin and azi- 
muthal motions agree or are opposed in direction. 
In its turn, however, this involves production of 
angular momentum about the vertical for which a 
couple must be applied by the frame, and of course 
to the frame by the operator. This couple is greater 
the greater Cn, and therefore if the operator cannot 
apply so great a couple, an azimuthal turning at 
rate w» cannot take place. With sufficiently great 
angular momentum the resistance to azimuthal turn- 
ing could be made for any stated values of 6 and o 
greater than any specified amount. 
The magnitude of this couple which measures the 
resistance to turning at a given rate is greatest when 
the angle @ is 90°; that is, when the axis of the 
flywheel is in the plane of the frame. 
Now I come to an interesting application of these 
ideas. You are aware that Lord Kelvin endeavoured 
to frame something like a kinetic theory of elasticity— 
that is, he conceived the idea that, for example, the 
rigidity of bodies, their elasticity of shape, depends 
on motions of the parts of the bodies, hidden from 
our ordinary senses, as the flvwheel of a gyrostat is 
hidden from our sight and touch by the case. Look 
at this diagram of a web (Fig. 11). It represents 
two sets of squares, one shown by full, the other by 
fine, lines; the former are supposed to be rigid 
squares, the latter flexible. Unlike ordinary fabrics, 
which are almost unstretchable except in a direction 
at 45° to the warp and woof, this web is equally 
stretchable in all directions. If the web is strained 
slightly by a small change of each flexible square into 
a rhombus, or into a not-square rectangle, the areas 
are to the first order of small quantities unaltered. 
Now imagine that a gyrostat is mounted in each of 
the rigid squares, so that the axis of the trunnions 
and the axis of rotation are in the plane of the square 
as shown in Fig. 12. If the angular speeds of the 
flywheels are sufficiently great, it is impossible to 
turn the squares in azimuth at any given small 
angular speed. Thus any strain involving turning of 
the small squares is resisted, and we have azimuthal 
rigidity conferred on the web by the gyrostats. There 
NO. 2366, VOL. 95] 
\ 

is, however, no resistance to non-rotational displace- 
ment of the squares as wholes. 
To get a model in three dimensions Lord Kelvin 
imagined an analogous structure made up of cubes, 
each composed of a rigid framework to play the part 
of the squares, and connected by flexible cords joining 
adjacent corners of the cubes. In each cube he sup- 
posed mounted three gyrostats with their trunnions 
at right angles to the three pairs of sides. This 
arrangement would, like the web of squares, resist 
[ ae ie 
a — 














Fic. 12. 


Fic. xr. 
rotation, but now about any axis whatever; and there 
would be no resistance to mere translation of the 
cubes as wholes. Thus the body so constituted would 
be undistinguishable from an ordinary elastic solid 
as regards translatory motion, but would resist 
turning. 
It is convenient in this connection to refer to an 
arrangement—a gyrostatic imitation of a spiral spring 
—in which a constant displacement is produced and 
maintained by the action of a constant force in a fixed 
direction, involving the application of a couple of 
constant moment, though not of constant direction 
of axis. This gyrostatic 
spring balance is indi- 
cated in a paper entitled 
“On a Gyrostatic 
Adynamic Constitution 
for Ether,” published 
partly in the Comptes 
rendus,* and partly in 
the Proceedings of the 
Royal Society of Edin- 
burgh.'° This is one 
of the many papers 
which Lord Kelvin pub- 
lished in the latter part 
of his life on a question 
that occupied him much 
from time to time, the 
nature of the ether as 
a vehicle of light and 
as the medium in which 


electric and magnetic ake 
phenomena are mani- ace 
fested. 
The spring balance is described in some detail in 
his ‘‘Popular Lectures and Addresses.” I had 
thought of realising the arrangement, which is shown 
in Fig. 13, but on consideration I found that though 
it would act as a spring, it would not, except under 
certain conditions, not easily realisable even approxi- 
mately, possess the peculiar property of a spiral spring 
of being drawn out a distance proportional to the — 
5 Comptes rendus, vol. cix., p. 453, 1889. Math. and Phys. Papers, vol. 
lii., p. 466. 
10 Proceedings of the Royal Society of Edinburgh, vol. xi., r8go. 
11 Vol. i., p. 237, ef segs 
