276 Proceedings of the Eoyal Society of Edinburgh. [Sess. 
Hence the axis of the gyroscope is upright after time Cn sin OJl. When 
00 is small the angular speed of recovery is given by IjCn, and the time of 
recovery by CnOJl. 
Again, let the gyroscope be deflected, with respect to the vertical, 
about the pivots P 2 P 2 ? ^^d let a couple of moment I, acting in a vertical 
plane containing i^ ^ parallel plane), be applied to the gyroscope. 
The effective couple about the pivots is I cos 0, where 0 is the 
deflection of the gyroscope on the pivots ^ 2^2 5 ^^d if the couple is an 
erecting one, we have 
dO I cos B 
dt i'jTi 
This equation may be written 
I _ do 
and its solution is 
, iCw , 
- A — loc 
^ I ' 
1 +sin 0 
1 - sin 0 
+ const. ; 
and if 0 = 0^ when ^ = 0, 
t 
log 
1 + sin Or. 1 
0 - lop; 
1 — sin 0Q 
1 + sin 0\ 
1 — sin 0/ * 
The time taken by the gyroscope to reach the vertical is thus 
^T 
log 
1 + sin 0Q 
1 - sin 0Q ’ 
which reduces to CnOJl when 0^ is small. Thus for small tilts the rates 
of recovery about the two axes are identical. It is usual in dealing with 
pivoted gyroscopic systems to assume that the arrangement of pivots 
shown in fig. 10 may be regarded as an ideal universal joint. 
If the constant couple I comes into existence when the deflection 
amounts to, say, four minutes of angle, then the resting position of the 
pivoted system will be definite within limits much closer than those 
required on an aeroplane for purposes of bombing. 
Suppose now the pivoted system set up on an aeroplane with the 
pivots p^p^ lying fore and aft. Let the axis of the gyroscope be initially 
upright, and suppose the aeroplane to move in a circular path. 
Let the horizontal acceleration which accompanies the turning motion 
of the aeroplane result in the establishment of the couple of moment I, the 
direction of the couple being such as to cause the gyroscope to turn, at any 
instant, on the pivots P 2 P 2 towards the apparent vertical. (It is to be 
