95 
1917—18.] Studies in Clocks and Time-keeping. 
thus for a = 2°, the coefficient would be less than (F’Ofi ; the same will apply 
to other cases ; nothing occurs to enhance any term which has not a 
period nearly equal to the principal period. In consequence, the sole 
sensible effect of the added terms will be found in the modification of the 
coefficients of (86). 
We come now to the inclusion of a maintenance term, R. 
First consider the effect of an irregular element in the term R, being 
an accidental variation, say, equal to AR. This will produce effects on both 
amplitude and epoch of motion, viz. by equation (10), 
Aa = — ■, I AR cos t dt , Ae = I a -1 AR sin r dt. 
n J n J 
These are Airy’s equations, discussed, however, only for the case in which 
k — 0. They were the first expressions given for the Escapement Error. 
They show, in the first place, that constancy of arc and constancy of epoch 
(or time-keeping) depend upon different or complementary conditions — this, 
of course, apart from the effect of arc upon time-keeping through the 
circular error. They show that the conditions for correct time-keeping is 
that Ae should enjoy a minimal property, vanishing for all small variations 
possible to AR — still apart from circular error. We shall return to their 
consideration later when the circumstances of regular maintenance have 
been dealt with. 
The mechanism of maintenance differs widely in detail in the clocks 
dealt with, but possesses certain necessary features in common. The 
pendulum first parts with energy in order to unlock the mechanism, and 
afterwards receives back from the mechanism a greater amount of energy. 
The excess is dissipated in overcoming frictional resistances, which increase 
with the amplitude of the motion and give an automatic balance, that would 
be maintained steadily if the circumstances did not vary. Effort is directed 
therefore to make the circumstances of unlocking and maintenance repeat 
themselves as exactly as possible both in force and in phase of application. 
As a matter of experience, the clocks of which I am speaking do remain 
in a sensibly steady state at least for several hours, or say for 10 4 semi- 
oscillations ; variations of a more fugitive character leave no feature that 
can be traced. Hence I shall consider such as covered by the accidental 
variations referred to under AR above, and I shall treat the maintenance 
and unlocking as constant, afterwards considering how far the coefficients 
that define them are themselves defined by the amplitude of the motion 
they maintain. 
It is obvious that the proper expression for a force (R), such as is 
