1921-22.] Gyroscope and “Vertical” Problem on Aircraft. 263 
reflection, that the gyroscopic pendulum will not take up the direction of 
this apparent gravity, but will process in such manner that (after a short 
interval of time) the rod r traces out a cone whose axis is oc and whose 
semi- vertical angle is tan-^a/^^. [In reality, when a state of steady pre- 
cession has been arrived at, the semi- vertical angle of the cone is slightly 
less (we suppose T' and Gn very great) than tan~^(a/^). This follows from 
the fact that the precessing system possesses kinetic energy, which kinetic 
energy is obtained at the expense of potential energy. Further, at the 
start of the motion gyroscopic oscillations of small amplitude are set up, 
and continue until destroyed by pivot friction.] 
The mass m moves in a circular path ahdea, the plane of which is 
inclined to the horizontal at an angle tan'^a/^. Since g is replaced by 
+ , the precessional period T' of steady precession is given by 
r 
mh J -f- 
Let the mass m occupy the position h (fig. 4) at time t. We have 
L ach 
and 
T' ’ 
Tvt 
r ' 
The angle made by the axis Om of the pivoted system with the true 
vertical at time t is the angle aoh. Hence, denoting this angle by 0, we 
have 
• 1 n 
sm = 
sin (3 sin 
irt 
r ’ 
when ^ = T74, sin Jd = sin /3 sin — ; when ^ = Ty2, sin |d==sin /3, or 0 = 2^] 
4 
when t = ST'/4), sin 10 = sin (3 sin fx ; and when t = T', 0 = 0. 
