106 
Proceedings of the Royal Society of Edinburgh. [Sess. 
We notice that this is independent of the angle y, representing the point 
where the crutch is picked up, and depends only on the angle S, represent- 
ing the surplus pressure which is the effective maintenance. S is given by 
comparing the angle between the stops with the whole arc. When the arc 
increases, S diminishes and so n increases, that is to say, the clock has a 
positive escapement error, tending to annul circular error, as is the case 
also with 2. and R. But in practice these theoretical errors are largely 
masked, as will appear hereafter. 
3. Riefler. 
The quantity H of p. 100 is the rate of supply of energy to the pendulum 
by the maintenance. As already explained, besides the deliberate supply by 
bending the spring, H should also cover the small loss occurring through 
friction with the scape-wheel tooth when the anchor is picked up at unlock- 
ing and the disturbance that may arise while the cock is free to rock upon 
its knife-edges. I shall, however, simply take the case where H is a given 
constant, C, from p = y to p = y + S, and again from = 77 - -f y to i r + y + d, 
and zero everywhere else, where 
p = nt + e 
and p vanishes when the pendulum is vertical. 
Then writing 
H = a 0 + a l cos p + \ sin p + . . . + a r cos p + b r sin p + 
we have 
7 TCL.j — C - 
| J 7 cos rpdp + j 
n+y+8 j 
cos rp d<p 
■+y J 
= 4Cr 1 cos r(y + 1-8) sin £rS 
- 0 . 
Similarly 
6,= C 
7 T() r 
X 
sin r<p d<p + 
r + y + 6 
TT + y 
sin r<p dp | 
= 4Cr 1 sin r(y + ^S) sin \rd 
- 0 , 
and 
so that 
27 tc«q = 2C8 ; 
[r even. 
[?’ odd. 
[r even, 
[r odd. 
again 
H = 2Ct t 8 + sin 8 {cos (2y + 8 ) cos 2 p + sin (2y + 8 ) sin 2 p [■ 
+ \ sin 28{cos (47 + 28) cos 4 p + sin (4y + 28) sin 4 p } 
+ . . . ] ; 
< 1 > = H tan p 
= C tan p from p = y to y + 8 and 
from p = 7r + y to 7r + y + 8 , 
