1921-22.] Gyroscope and “Vertical” Problem on Aircraft. 265 
Let, as before, the periodic time of steady precession be denoted by T, 
and suppose that the time of the aeroplane in the circle is T'. Let the 
aeroplane be at P at time t, counting time from the instant corresponding 
27t 
to B. The angle BOP is 7 ^^ (fig. 6 ). (This figure is intended to show 
the pivoted system at the instants corresponding to B, P, and Pj. It will 
be readily understood that the scale of the diagram cannot be near the 
true scale. As a rule the radius of the curve in which a large bombing 
plane turns is about 200-250 yards.) The centrewards acceleration of the 
aeroplane is , where v is the linear 
speed of the aeroplane. At any instant 
the force acting on the mass m due to the 
centrewards acceleration of the plane is 
27T 
, and this force turns in azimuth 
with the aeroplane. The couple experi- 
27T 
enced by the pivoted system is ihvy^Ii , 
and the angular speed of the precessional 
motion at the instant is this couple 
divided by the angular momentum of the 
gyroscope. Now at time t the aeroplane 
is at P, and the pendulum is turning 
instantaneously about the pivots , 
which at the instant are parallel to OP. 
When the plane arrives at P^ the pivots are parallel, and the pivots 
V 2 P 2 perpendicular, to OP^ . Denoting the inclinations of the pivoted system 
to the vertical at time t, about axes parallel and perpendicular to OP^ , by 
G and 02 > respectively, we have, with ijj the angle so marked in fig. 6, 
2- 
-h 
9- 
Fig. 6. 
dt 
and 
dO 
dt 
2 _ 
Cn 
27T 
mvT^h 
T' 
Cn 
The first of these equations gives, since mhlCn = 27rl{gT), where T 
the period of the pendulum for steady precessional motion, 
2tt, 
IS 
n 27TV ■ 
c'l = — — sm 
•A 
T' 
t + const. 
