302 Proceedings of the Koyal Society of Edinburgh. [Sess. 
The angle turned through in the time tt/co, during which the couple acts, 
is 2mg{d—2r)lCnoo. 
This result may also be obtained as follows. The ball applies, at each 
instant, a couple of amount mgr, about 0. When it is in contact with its 
pusher the system is balanced. If the ball were removed from the track 
there would appear a couple of amount mgr, which couple would rotate 
at angular speed w. The time taken by the ball to travel a distance d — 2r 
(its maximum shift relative to the pusher and check) is (d—2r)lr(jo. When 
the angular displacement from od (fig. 27) is /5 the ball is suddenly 
transferred from the pusher to the check ; thus it virtually disappears 
from the track for the above interval of time. When the angular displace- 
ment of the ball is /5 4- tt it spends time 2(d — 2r)/rco on the track ; it goes 
through the distance d — 2r in contact with the check, and then, after its 
retransference to the pusher, over the same distance in contact with the 
latter. Thus in a complete turn of the erector the integral effect is to 
take off the ball, when the angular displacement is ^ for time {d — 2r)jrw , 
and to place it on the opposite side of the track for the same interval of 
time. Clearly this results in a couple of amount 2mgr acting about 
an axis perpendicular to dod' for time {d—2r)!roo. The angle turned 
through, by the pivoted system about an axis parallel to dod' , is thus 
2mgr(d~2r)lrcoCn or 2mg{d — 2r)lo)Cn, as before. 
In time T', the periodic time of the spider on which mounted the 
pusher and check, the system turns, under the action of the integral 
couple due to the transference of the ball from the pusher to the check, 
and from the check to the pusher, through an angle 2mg{d—2r)looCn, or 
2mgalcoCn, where a = d — 2r. Since the integral couple occurs once in a 
revolution of the erector, and (o= 27 t/T', the mean rate at which the pivoted 
system turns about an axis parallel to dod' is mgalirCn. If two balls are 
employed, each operated by a pusher and check, situated at opposite ends 
of a diameter of the erector, the average rate of turning is 2mgal'rrCn. 
For three balls, symmetrically distributed, the rate of turning is SmgalirCn ; 
and so on. 
Let there be two balls operated as described. If each ball is trans- 
ferred from the pusher to the check when its angular displacement from 
the line of greatest slope is then, as already explained, the pivoted 
system turns about an axis parallel to dod' at angular speed 2mgalirCn. 
Denoting the rate of turning about the axes P 2 P 2 (hg- 2*^)^ the 
athwart and fore-and-aft pivots, by and 6^ respectively, we have 
A 2mqao,os, B a 2mqa^\r].B 
o • 
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