1921-22.] Gyroscope and “ Vertical” Problem on Aircraft. 283 
a half turn, = 0 and = mg{r^ — r.^. After a quarter turn, = mg{r^ — r^} 
and Cg = 0. After a half turn, — mg{r^ — and = 0. During the half 
turn executed at uniform angular speed, increases from 0 to +mp(r^ — rg) 
and then diminishes to zero ; at the same time diminishes from mg(g\ — r^ 
to 0 and increases, with its direction changed, to mg(r-^ — r^). Thus, after 
the completion of the half turn the pivoted system has turned wholly 
about the pivots P 2 JP 2 ' And since the action is continuous the erector 
applies to the gyroscope an integral couple which causes it to approach the 
vertical wholly about the pivots PgPg • 
With the balls in the positions shown in fig. L5, we have for the couple 
applied at the instant, about the pivots mg{r^ — and for the 
angular speed at which the pivoted system is approaching the vertical on 
the pivots P 2 P 2 , — rg) sin /3/C?i, where Cn is the angular momentum 
of the gyroscope. Counting time from the instant at which the ball 6^ 
was at a, and denoting the angular speed of the erector by m, we have 
cl9 sin 
(it C?^ ’ 
where 6 is the inclination of the pivoted system (in the pivots p)-iP 2 ) ^be 
vertical at time t. This equation gives 
6 = — f?) cos (Jit + const. 
Qlld) 
Let 0 = 00 at time = 0, and we have 
Liridi 
This equation holds for any instant during the half turn. The ball 
arrives at c and the ball at /, after time tt/w. If the inclination of the 
pivoted system to the vertical is then G » we have, since cos has then 
the value — 1, 
. „ - j-j) 
Now 0o“^i ^be angle through which the system has turned towards 
the vertical in time T72, where T' is the periodic time of the erector. The 
average angular speed of erection is thus (0 q — G)/(Ty2), and since a) = 27r/T', 
- r^) 
T'12 TrCn 
In the erector as shown in the figures two balls are employed. If 
the plate i is provided with three slots, each surrounding a ball, the rate 
of recovery is Smg(o\ — r^)l 7 rCn. If n balls are employed, the rate is 
nmg{r-^ — r^jTrOn. 
