1917 - 18 .] Studies in Clocks and Time-keeping. 
169 
XVII. — Studies in Clocks and Time-keeping : No. 2. Tables of 
the Circular Equation. By Professor R. A. Sampson, F.R.S. 
(MS. received and read June 3, 1918.) 
The period of oscillation of a free pendulum increases with the arc of 
oscillation. This effect is known technically as the Circular Error. Its 
theoretical amount for the simple pendulum is readily calculated, and the 
following tables give it. I have called it the Circular Equation in place of 
the Circular Error, employing the word equation in its astronomical mean- 
ing of a theoretical numerical adjustment of a result, and distinguishing 
it thus from discrepancies of unknown source and amount which may 
also present themselves. The relation between arc and rate in the 
maintained motion of the pendulum of a clock will be a subject for exam- 
ination experimentally and otherwise in future numbers of these studies. 
It may contain several elements, but must include the Circular Equation 
among them. 
There are two tables. The first shows the value in terms of true 
seconds of the period of a complete semi-oscillation for all semi-arcs of 
oscillation up to 300', and the consequent losing rate per day of a pen- 
dulum which would swing true seconds for zero arc. The second table 
shows for the same series of arcs the time occupied to swing any portion 
of those arcs, first in terms of its own seconds and then in phase angle, 
taking 90° to the quarter oscillation. 
The following examples illustrate the use of the tables : — 
I. What is the period for a semi-arc of 120' of a pendulum which 
swings true seconds for zero arc ? 
Entering Table I. with argument a = 120', we see that it is 
1*0000 7616. 
II. If the semi-arc of oscillation increases from 100' to 101 ', what is 
the consequent loss in rate per day ? 
From Table I, the daily rate relative to the simple pendulum 
with zero arc is 4 S, 570 for 100' and 4 S *662 for 101'; the difference 
gives the answer to the question, viz. 0 S, 092. 
III. If the free oscillation of the pendulum (“ supplemental arc ”) begins 
at 60', and the whole semi-arc is 150', what fraction of time 
does the supplemental arc occupy? 
