1921-22.] Gryroscope and “ Vertical” Problem on Aircraft. 285 
the speed at which the balls travel on the plate. This will be at once 
apparent if the number of balls employed is supposed very great. 
The rate of recovery of the device is proportional to m, the mass of a 
ball, to the number of balls, and to — the rate is inversely propor- 
tional to Cn, the angular momentum of the gyroscope. 
Although the rate of recovery of the device is independent of the erector 
speed, as has been shown, the action does depend on the speed. In order 
that the erector may function after the manner described, it is necessary 
that the balls, when they arrive at c and / respectively (fig. 15), should roll 
down the slope into the inner and outer compartments of their containing 
slots. Consider the ball 5^ when at c. It is resting on a plane which is 
sloped, along co, at an angle d to the horizontal. Further, it is accelerated 
towards o, the centre of the path, the amount of the acceleration being 
The condition that the ball should press against the back of the slot 
is clearly that should be greater than g sin d. Consequently, unless 
g sin d is greater than wVj the erector will cease to function. Moreover, 
supposing g sin d to be greater than , the acceleration of the ball, 
relative to the plate Ag > ^ ~ 9;nd this acceleration must be 
sufficiently great to permit of the ball travelling from c to in a short 
interval of time. 
In practice, of course, the radial motion of the ball starts before the slot 
arrives at the highest position in its path, and in point of fact the path of 
each ball on the plate h is not strictly that shown in fig. 16. At an instant 
at which the line joining the centre of the plate h to the point of contact of 
the ball with the plate makes an angle 0 with the line of greatest slope, 
oc, the acceleration of the ball relative to the erector is sin d cos ^ — wV), 
where r is the distance of the centre of the ball from the centre of the 
erector. As the ball ascends, the angle 0 diminishes and the acceleration 
increases to the value ^{g sin 6 — wV) and then diminishes. Again, once the 
radial motion of the ball starts, r diminishes and the acceleration of the 
ball increases. 
The line joining the centre of the plate to the point of contact of the 
ball with the plate advances from the position in which its angular dis- 
placement to the left of oc is (p, to one in which its angular displacement 
to the right of oc is (p in time 2(plw. Hence it is clear that the ball will 
arrive at the inner compartment of its surrounding slot provided that 
?‘i = ^ ^{g sin 6> cos </) - co%)^ . 
For an erector of this type in which ^^ = 3 inches, — inch, w = ^ 
revolutions per minute, the limiting value of d is about A of ^ degree. 
