160 
Proceedings of the Royal Society of Edinburgh. [Sess. 
We have here a rather complicated dependence on o-, p, and Ic. To make 
the matter more definite, we will suppose the damping to be such that the 
oscillations of the frame about its horizontal axis are on the verge of 
aperiodicity when there is no rotation, so that k — 2p. We have then, 
in place of (37), the expression 
27tC%2 2. 
at 
ipa- 
(cT^+P^y 
(38) 
For a given value of cr this is a maximum for maximum value 
being 
3J37^C27^2 
4AIo-‘^ 
(39) 
As a numerical example suppose that we have a vessel of 1000 metric 
tons, whose metacentric height is 50 cm., and that its free period of rolling 
is 10 secs., so that the moment of inertia about a longitudinal axis is 
I = 1-240 X 10^ kg. ml For the moments of inertia of the gyrostat we will 
take A = C = 1500 kg. ml, and for the speed of revolution n = lb0 sec."^ 
If we put 27r/(T=10, the consequent value of the expression in (39) is 28 ! 
The suppositions we have made are in some respects extreme, but it is 
evident that even when the above conditions are only imperfectly realised, 
the forced oscillations of the vessel, due to the action of waves, may be 
enormously reduced in amplitude, whilst free oscillations are powerfully 
damped. 
Returning to equations (30) and (33) we note that 
per k 
■ (40) 
In real form, the path of the pole of the fiywheel is given by the equations 
X = - sin {a-t + T) , y = y^QOSat . . . (41) 
k 
* Round numbers have been taken, but the order of magnitude is in each respect the 
same as in a practical example given by Klein and Sommerfeld, p. 832. 
