1917-18.] Studies in Clocks and Time-keeping 
Then we have 
L r/A — f si 
Jo 
103 
7rA~ 
7TT 
^ + 8 . 7t(t _ tj-\ 
Sill cos vtcIt - I Sill v 
o 
6 
cos IttlIt 
'8 7rr fS _ ?rT 
= I sin — cos vt'It - | sin — cos (?*7r + ?*t)cZt 
'0 
,8 
= 0, 
similarly 
= — cos 2 4?'8 . 47rS/(r 2 8 2 — 7r 2 ) ; 
7rJB,/A = 0, 
[?• even. 
- **) ; 
[r odd. 
[r even, 
[r odd. 
for the special case rS = ir, these reduce respectively to 0 and ir/r. Thus 
R 
4A8 
7T~ 
cos (r - ^8) cos |8 j N -J + cos 3(t - ^8) cos 48 / [ 1 
3 2 8 2 
71" 
the first term, which becomes much the most important, is marked by the 
dotted line in the figure ; it will be remarked that its general character 
could have been sketched in without ambiguity from a consideration of 
the value of R, especially in the important respect of the position of the 
maximum. For example, if we add to the maintenance here shown the 
force on the pendulum required for unlocking, which will be represented 
by a small addition preceding the maintenance, and of the contrary sign, 
this would shift onwards the position of the maximum. 
The resulting- value of x is 
x 
4A8 
7 r 2 
//- 
S 2 
sin r . (xn) 1 cos 2 18 j y 1 cos -f-8 cos 3(r - ^8) 
f 
1 - 
3 2 S 2 
( 21 ) 
where 
— = { 1 + ~ tan 2 48 J- - -A tan 48 
n t n 2 1 J 2 n 
In this expression we remark that 
cfis 2 18/(1 - S 2 /tt 2 ) = (1 - S 2 /3 2 7r 2 )f 1 - S 2 /5 2 tt 2 ) . . . 
which we can generally treat as equal to 1, writing thus 
x= 4A8(7 t 2 k7i)~ 1 sin r+ . . . 
These are the equations showing the amplitude and period under the 
given maintenance, which may be taken as representative of the class in 
which pressure is exerted on tlie pendulum after it has passed its vertical 
position. We see that if the arc increases owing to a decrease of k, n will 
increase or the clock will gain, so that the escapement error will tend to 
counteract the circular error. 
It is easy to see that if the same maintenance begins not at r = 0 but 
at r = y, lasting to r = y + d, the formula will run 
n 
7 ii = i + 
2 n 
tan (y + 18) 
. (21a) 
