286 Proceedings of the Koyal Society of Edinburgh. [Sess. 
When inclined to the vertical the pivoted system recovers, up to this 
limiting value of without processional motion in the ordinary sense of 
the term. That is, if inclined on the pivots it approaches the vertical 
on the pivots ^’^d similarly for any other axis. The recovery is not 
accompanied by spiral motion of the axis of the gyroscope. 
To render the device delicate the slots are so contrived that the balls 
possess a certain amount of freedom when situated within the outside com- 
partments of the slots. When the limiting value of 6 has been reached, 
and the radial motion of the balls has ceased, they move to and fro in the 
outer compartments of the slots, and as a result the pivoted system is 
erected into the vertical with great exactness. The resting position of the 
system, with this simple form of erector, is exact to within 3 or 4 minutes 
of angle. 
The dynamical action of the erector during this final stage of recovery 
is important, and will be returned to later. 
An examination of fig. 14 (h) will show that the form there given to 
the slots is defective in that the centres of the balls do not lie on a line 
which passes through the centre of the plate i. When situated as shown in 
the figure the balls apply a couple about the SbX.\s p^p 2 , and this causes the 
system to process about the pivots With the slots formed as shown, 
after a half turn of the erector the effect is to turn the pivoted system 
about Pi2^i, and as a result the recovery of the device is not effected 
wholly about P 2 P 2 ' 
In fig. 17 is shown a later construction given to the plate i. In the 
figure the pivoted system is supposed inclined on the pivots p.^p^, the erector 
towards the reader, so that the erector is functioning. It will be seen that 
the centres and of the balls lie on a line which passes through the 
centre o of the plate. Further, it will be seen that the outside boundary 
lines of the slots are curved towards the centre of the plate. This has 
the effect of constraining the balls, as a consequence of their centrewards 
acceleration, to arrange themselves on the plate h, when the pivoted system 
is upright, so that they rotate as an approximately balanced system. In 
order that the delicacy of the arrangement should not be diminished, the 
boundary of each slot, in the neighbourhood of the equilibrium position of 
the enclosed ball, is cut so as to form a portion of a circle with o as centre. 
The resting positions of the balls are not quite definite, as a result ; and as 
a consequence the pivoted system when upright, in general, performs oscil- 
lations of very small amplitude about each of the pivot axes p^Pi and P 2 P 2 ' 
This is an advantage, in fact, as the pivots are maintained rocking to and 
fro in their bearings, and pivot friction is thus reduced. 
