278 Proceedings of the Royal Society of Edinburgh. [Sess. 
about the axis parallel to dh, and when ci passes to the other side of ca 
the direction of the second component couple changes sign. During the 
first half of the interval of time corresponding to if; the gyroscope is 
turning about an axis parallel to db and also about an axis perpendicular 
to db. The same applies to any instant during the second half of the 
interval ; but whereas the turning about the former axis is in the same 
direction as before, that about the latter axis has been reversed. After 
the aeroplane has turned through the angle ip the gyroscope has been 
deflected wholly about an axis parallel to db. The total deflection 6 is 
given by 
( 9 = 
\f/Ln 
If and 6^ denote the inclinations of the gyroscope about the pivots 
and P 2 P 2 respectively, we have 
The aeroplane turns for time ip/ip, and after that time the gyroscope 
is inclined to the true vertical at an angle 21 sin ^iplipCn. If now the 
aeroplane resumes straight flight, the pivoted system approaches the true 
vertical at angular speed l/Cn, and is upright in time 2 sin ^ip/^, or 
2t sin \ipjxp, where t is the time taken by the aeroplane to turn through 
the angle ip. 
For small values of ip, as for example the turns brought about by 
yawing motion of the aeroplane, the time required for correction of a 
deviation from the vertical is equal to the time spent in the turn. If the 
aeroplane turns through 90° in time t, and thereafter moves in a straight 
path, the deviation introduced during the turn is corrected in time 
7 T. If Ip is 180° — that is, if the aeroplane executes a half turn — the 
resulting error is corrected in time 2^/7 t, where t is now the time in the 
half turn. Thus, by operating the gyroscopic system in the manner 
described, the disadvantages which are inherent in gyroscopic pendulums 
depending for their action on the provision of gravity control and viscous 
damping are completely eliminated. 
In calculating the data set out in the following table the angular 
momentum of the gyroscope is taken as 250, in foot, pound, second units, 
^ as ^ of a poundal at an arm of I foot, and the aeroplane is supposed to 
turn at an angular speed of 9° per second (in 1917 it was usual for a 
bombing plane, just before running up to a target, to execute a half turn, 
and the half turn was executed in about 20 seconds). After turning 
