1921-22.] Gyroscope and “Vertical” Problem on Aircraft. 299' 
Otherwise the construction is as before. The pivoted system is attached,, 
as before, at to a frame /, and this is in turn pivoted at P 2 P 2 uprights 
carried by a base as shown. When the pivoted system is upright the track 
in which the balls revolve is horizontal. 
Let now the pivoted system be inclined to the vertical. The track i& 
inclined to the horizontal, and each ball in ascending the slope of the track 
remains against its pusher ; but when it crosses the crest of the slope, and 
descends, it is situated on a slope down which it is accelerated. Thus it 
runs ahead of its pusher. After passing the lowest point of the track it 
ascends the slope, its motion is retarded, and it again comes in contact 
with its pusher, against which it presses until it next crosses the crest 
of the slope. Hence each ball spends more time in ascending than in 
descending the slope of the circular track in which it is constrained to 
move, and the result is the application to the pivoted system of an 
integral couple which causes the inclination of the system to the vertical 
to diminish, and finally to become zero, or at all events to arrive at a 
minimum value. 
It is proposed now, on account of the extreme practical value of this 
type of erector, to investigate the nature and amount of the integral 
couple obtained, and how the couple depends on the slope of the track to 
the horizontal, the masses of the balls, the travel of the balls relative to 
the pushers, and the speed of the erector. 
Fig. 27 shows a single ball h which is propelled on a fixed circular 
track ^ by a pusher p. Its travel on the track, relative to the pusher, is 
limited by the check c. The arrangement of ball pusher and check is 
shown in elevation in fig. 27a. The direction of rotation of the ball in 
the track is supposed to be clockwise, as viewed from the upper side of 
the track. 
Let m be the mass, and r the radius of the ball, d the distance between 
the pusher and check, and suppose that the track is inclined at an angle d 
to the horizontal about the pivots p^P^. (bg- 27) in the clockwise direction 
as seen from the left-hand side of the diagram, so that d is the crest of 
the slope. Let the pivoted system be neutral with respect to the pivots 
ViVi ’ P 2 P 2 (%• 27) when the ball is in contact with its pusher. 
The ball, when in the position d, has just ascended the slope, and is 
in contact with its pusher. At the instant, since it is at the crest of the 
slope, it is subject to no accelerating force. As it moves round it enters 
the slope, is accelerated, and comes against its check. It now descends 
and remains in contact with the check until it is again on the ascending 
side of the track, when its motion is retarded, with the result that it is 
