80 
Proceedings of the Royal Society of Edinburgh. [Sess. 
One striking feature of the maintenance just described is the automatic 
compensation it supplies for excessive or defective arc. If for any reason 
the arc is above its mean value, the pendulum carries the disc more rapidly 
away from contact with the impulse piece, and less work is in consequence 
done upon the pendulum, because they part company at an earlier point. 
So that an excessive arc receives a share of work that is below the mean. 
This is an important feature. It is true that every pendulum possesses in 
some degree such an automatic compensation in view of the greater pro- 
portionate resistance of the air when the arc increases, but its operation is 
nothing like so prompt and decisive. It deserves a short discussion on its 
theoretical side. 
The impulse piece acts upon the roller which is pivoted in the crutch of 
the pendulum. This roller is at rest when the impulse piece first descends 
upon it, but the impulse piece exerts a frictional drag upon it which almost 
at once establishes rolling contact between them. It may be shown by 
writing down the equations that this rolling contact always ceases at a 
point shortly before the pendulum and impulse piece part company, and is 
replaced by sliding contact. The effect of the frictional drag is pro tanto 
to hold back the pendulum, but energy so lost is in large degree merely 
stored in the spin conferred upon the roller, which returns part of it to the 
pendulum through action of its pivots. I shall not write down equations 
for these intricacies, but shall confine myself to the action in three simplified 
cases — viz. (1), when the circular roller is treated as a fixed pallet of 
circular cross-section and friction is neglected ; (2) when the fixed pallet is 
of triangular cross-section and friction is neglected — a case that could be 
approximately realised by attaching a small roller to the impulse piece ; 
and (3), the second case as modified by friction. In all, the impulse piece 
is treated as descending freely between vertical guides and the pallet as 
moved away transversely by the pendulum, with constant velocity. 
Case 1. — Smooth Circular Pallet. 
Let m be the mass and f the acceleration of the impulse piece when 
unresisted. Q the normal force between it and the pallet, a the radius of 
the pallet, 0 the angle this radius makes with the vertical in any position ; 
then if V is the velocity of the pendulum at this point, we have during 
contact 
Y = d(a sin 0)/dt — a cos 0 . O', 
and the actual acceleration of the impulse piece is 
- d 2 (a cos 0)ldt 2 = V 2 /a cos 3 0 ; 
