1921-22.] Gyroscope and “Vertical” Problem on Aircraft. 267 
direction of the line which bisects the angle between the initial and final 
directions of the pivots ^ 2 ^ 2 ? have 
0 = - 
m vli . 
Cn 
sin + const. 
When ^ = 0, 6 = 0, so that the constant is mvh sin ^ipjCn. 
Hence 
. tuvIl r . 1 , • / 1 , 27t 'I 
^ ^ I sin h'l' - sm 
r 7/ 
When the aeroplane has turned through the angle i/j we have ifj = 2iTtjT' , 
and hence then, 
Cti 
' 1 t 
= ^ 2 sin|i/. . 
Thus when the aeroplane has turned through an angle 0 the pendulum 
is deflected, with respect to the vertical, about a horizontal axis which 
bisects the angle between the initial and final horizontal projections of the 
pivots Denoting as before the deliections of the pivoted system 
about the pivots and P 2 P 2 by G and 63 ? we have 
2tTTV 
6 -^ — cos 
2itV 
w 
sin if, ; 
sill-1 M 
2 yy 2 r 
2ttv 
(1 - cos if/) . 
As an example, we suppose the pivoted system to have a periodic time, 
for steady motion, of 6 minutes, and the speed of the aeroplane to be 
100 feet per second. The aeroplane is supposed to turn in a circular path, 
and the pivoted system to be initially upright. We have 
^ 2tt X 100 X 5/ 3j-v 'll c OK ■ 1 / /■ j ^ 
d = — — — - — 1 sm = 6* Jo sin ^if/ (in degrees) 
oZ X o o u 
^j = 3T2sini// (in degrees), '^'12(1 — cos if/) (in degrees). 
