270 
Proceedings of the Koyal Society of Edinburgh. [Sess. 
gyroscope, with its attachments, is mounted within the frame / so that 
the centre of gravity of the system lies considerably below the plane of 
the pivot axes. 
The action is as follows. When the pivoted system is inclined to the 
vertical, precession takes place, with the result that, as already explained, 
the axis of spin moves round so as to sweep out a cone. If, for example, 
at an instant the inclination is about the pivots P 2 V 2 ’ ^he system is turning 
instantaneously about the pivots Now consider the reservoirs which 
lie transverse to the pivots 'P 2 V 2 • ^f these is higher than the other 
and oil moves down the slope, so that it collects in the lower bottle. As 
time proceeds and the tilt moves round as a result of the precessional 
motion, it follows that the gyroscopic system is called upon to raise the 
heavier bottle. Thus work is being continually done by the gyroscopic 
system, which work is performed at the expense of the potential energy 
possessed by the system. This energy is of course dissipated in heat within 
the dashpot. 
It will be clear that when the tilt becomes small the action of the 
dashpot becomes feeble. As the tilt becomes smaller the rate of dissipation 
of energy in the dashpot diminishes, and, with the construction described, 
will cease before the true vertical is attained. This follows from the fact 
that the oil must determine, to some extent, the resting position of the 
pivoted system, inasmuch as the resting position depends on the distribu- 
tion of the mass of the entire pivoted system. 
Clearly the rate of dissipation of energy" in the dashpot, for a given 
tilt, diminishes as the periodic time of the pivoted system is increased. 
When the period is great the pivoted system, when disturbed, returns very 
slowly towards the true vertical ; and also the resting position is seriously 
affected. When the period is small the disturbances introduced by turning 
motions of the aeroplane are very great. To obtain a quantitative idea 
of the mechanics of viscous damping, as applied to gyroscopic pendulums 
contrived with the object of defining the true vertical on aircraft, consider 
again the device shown in fig. 7. Let it be supposed set up on a table and 
deflected with respect to the vertical. It precesses, after the manner already 
described, in periodic time T given by 
T = 277-*^'* 
mgh 
The angular speed of the rod r about the intersection of the pivot axes 
is ^ sin 0, where 0 is the inclination of r to the vertical. It is character- 
istic of viscous resistance, for the speeds with which we are here concerned, 
