1921-22.] Grjroscope and “Vertical” Problem on Aircraft. 301 
the ball (from the check to the pusher) the centre of gravity of the system 
is once more at the intersection of the pivot axes, and it remains so until 
the pusher and ball arrive once more in the positions of p, h', when the 
couple reappears, to advance through an angle tt ; it then disappears ; 
and so on. 
The couple may be represented completely by a line a, of appropriate 
length, drawn perpendicular to the plane in which it acts at each instant. 
This line, which represents the couple, rotates at the angular speed w of 
the erector. It appears at the instant corresponding to the angular dis- 
placement /3 (fig. 27), disappears at the instant corresponding to /B + ir; 
and so on. 
During the first half turn of the couple the integral effect has clearly 
been to turn the pivoted system 
about an axis parallel to dod' (fig. 
27). From symmetry it follows 
that the integral couple is com- 
pletely represented by a line 
drawn perpendicular to dod'. 
This couple is applied about an 
axis perpendicular to dod', and 
the gyroscope of course turns 
about an axis parallel to dod'. 
When the erector turns in the 
direction of the spin of the gyro- 
scope, the pivoted system turns, as a consequence of the applied couple, in 
the direction which results in the inclination of the pivoted system being 
diminished. Before the application of the integral couple the inclination 
of the pivoted system was wholly about After the application of 
the couple the inclination has been diminished and the line of greatest 
slope (fig. 27) has turned in the clockwise direction through a small angle. 
This follows from the fact that the effect of the integral couple has been to 
turn the pivoted system on the pivots about an axis parallel to dod' , and 
not wholly on the pivots P 2 P 2 • 
To obtain the angle through which the pivoted system has turned, we 
count time from the instant at which the first transference took place (the 
transference is supposed sudden). At time t the couple has turned through 
an angle wt, and its component about an axis perpendicular to dod' is 
mg{d — 2r) oot (see fig. 27a). The rate of turning of the pivoted 
system about the pivots under the action of this component couple is 
mg{d — 2r)^\nwtlCn, where Cn is the angular momentum of the gyroscope. 
