292 
Proceedings of the Eoyal Society of Edinburgh. [Sess. 
azimuthal turning is reversed, so also is the direction of the applied 
gyroscopic couple and the direction of turning of the pivoted system 
on 
Let now the apparatus be supposed set up on an aeroplane with the 
pivots P 2 P 2 and so that the spin of the gyroscope g' is 
counter-clockwise as seen from the rear of the cockpit. Starting with 
the apparatus upright, let the aeroplane turn in azimuth at angular 
speed i[i. In consequence of the couple due to the forcible turning of g\ 
the pivoted system turns on P 2 P 2 away from the true vertical, and so as 
to increase its inclination to the apparent vertical. 
Let the pivoted system include an erector of the type illustrated in 
fig. 12. When the aeroplane changes direction the erector turns the 
pivoted system on P 2 P 2 towards the apparent vertical at angular speed 
2mg(rj — r^)l 7 rCn. At such instant the couple due to the turning of in 
azimuth causes the system to turn on p^p^ fhe opposite direction, so that 
the net rate of turning towards the apparent vertical is given by 
^ 2mg{r-^ - r^) loji/)- ^ 
rrCn Cn 
The system behaves as a neutral gyroscope if 6 = 0] that is, if 
2mg{rj^ — r 2 ) = 7rl(joi[f. If, therefore, the value of Ico be so chosen that 
this equation is fulfilled for the value of ip most generally adopted in 
practice, the apparatus becomes endowed with the property that when the 
aeroplane is turning the pivoted system can move away from the true 
vertical towards the apparent vertical only very slowly, hut when the 
curved motion ceases it returns to the true vertical relatively quicJdy. 
Let the speed of recovery due to the erector be, say, 1 degree in 20 
seconds of time, and f he 9 degrees in 1 second of time. If the angular 
momentum of the gyroscope g is 250, in foot, pound, second units, we 
have 
250 
Io) = ^^^ — ^ = 1*4 (approximately, in foot, pound, second units), 
which shows that the small gyroscope may conveniently take the form 
of a flywheel attached to the spindle of the turbine employed to rotate 
the erector (fig. 12). 
It has been shown that the ball erector as described causes the pivoted 
system, when the system is inclined, to approach the true vertical at an 
angular speed which is independent of the speed at which the erector 
rotates, and that the radial motion of the balls, on which the erecting 
action depends, ceases when o)Vi>^sind. Hence, if, when the device is 
