1921-22.] Glyroscope and “ Vertical” Problem on Aircraft. 277 
clearly understood that the gyroscope g is supposed to be in neutral 
equilibrium with respect to the pivots p^Pi and P 2 P 2 > that the turning 
of the gyroscope on the pivots is due solely to the action of the couple of 
moment 1.) The total deflection of the axis of the gyroscope from the true 
vertical, after the aeroplane has turned in azimuth through an angle ijj, is 
easily obtained. In fig. 11, (a) shows the pivoted system in the initial 
position, and (b) the pivoted system in the final position, after the aero- 
plane has turned through the angle ijj. The following is the approximate 
solution. 
The couple acting at any instant (of constant moment 1) may be repre- 
sented by a line drawn perpendicular to the vertical plane containing P 2 P 2 
at the instant. This line turns round with the aeroplane at angular speed 
ijj, and describes a total angle Let ac (fig. 11) bisect the angle between 
the initial and final directions of the vector which represents the couple. 
At time t the couple is represented by ci, and is equivalent to two couples 
represented by ch and hi respectively. The first of these causes the 
gyroscope to turn about an axis parallel to db, where db is perpendicular 
to C(X, and the latter causes the gyroscope to turn about an axis per- 
pendicular to db. • When ci lies along ca the gyroscope is turning wholly 
