1921-22.] Gyroscope and “ Vertical ” Problem on Aircraft. 303 
In actual practice the behaviour of the balls propelled on a track, and 
controlled by pushers and checks, after the manner described, approximates 
closely to that explained above. 
Now, when a ball is in the position corresponding to the angular dis- 
placement ^ its acceleration along the plane is sin 6 sin /3 (for the 
pivoted system is supposed inclined to the vertical at an angle 6 on the 
pivots ^> 2 ^ 2 )- ^ small, clearly for large values of 6 the transference 
of the ball from pusher to check will take place before ^ is great. If 6 is 
small, P will be relatively great. Now if /3 is small, G is great compared 
with 62 ; when /3 is great, 6 ^ is great compared with G • 
Let now the device as described be mounted on an aeroplane with 
the pivots P 2 P 2 iyi^& ^^d aft. When the aeroplane turns, the pivoted 
system, supposed upright initially, turns on the pivots P 2 P 2 towards the 
apparent vertical, and on the pivots p^Pi in a direction depending on the 
direction of turning of the plane. If v is the speed of the aeroplane, and ijj 
is its azimuthal rate of turning, g is replaced by and the slope 
is tsin~^{vifjlg). Since during manoeuvres of aeroplanes i[j is usually, and v 
is always, great, it follows that /3 will be small, and hence that initially 
will be great and small. If a small weight is attached to the pivoted 
system in line with the axis, so as to provide the stabiliser with a small 
amount of gravity control, this weight acts in opposition to the couple 
which is bringing about the turning 0 , about the pivots PiPi . If the mass 
of this attached weight is m and its distance below the intersection of the 
pivot axes is A, the couple it applies about the pivots p> 2 P 2 ’ ^^^d which 
causes turning about the pivots p^p^, is mvijjh. Hence, with this addition, 
Oy will be zero if 
77l'v\ph 
2mga cos p 
7T 
As already pointed out, for an aeroplane v is always (and ip is usually) 
great. Provided that the speed of the erector is small, the transference of 
a ball from its pusher to its check will be accomplished for a small value 
of /3. Now 62 is small when /3 is small. Hence with a proper choice of 
o) and of m the device may be rendered blind to the apparent vertical. 
When straight flying is resumed, following a turn, any slight deviation 
of the pivoted system will be corrected about an axis perpendicular to the 
vertical plane which contains the axis of the pivoted system. This follows 
from the fact that when /3 approaches 90°, 6 ^ is small and relatively 
great. 
As an example, consider the case of an erector, of the type described, 
functioning on an aeroplane. Let m=l ounce, u = I inch, C'TI = 250, in 
