1917-18.] Studies in Clocks and Time-keeping. 
105 
2. COTTINGHAM. 
The crutch is picked up on the swing to the right, say, at r = y, and 
dropped on the return swing at r = ir — y + S. During the motion crutch 
and pendulum are one piece, as nearly as can be made ; but regarding the 
pendulum alone, the maintenance may be taken as a constant pressure 
upon it, always in the same sense, as shown in fig. 7. The point at 
which the crutch is dropped may or may not be beyond the point B' on the 
return swing. The crutch is picked up while the pendulum has consider- 
able velocity, and there is a loss of energy owing to the impulse between 
them. This is the only parallel to “ unlocking ” that this clock presents. 
The loss due to this, and due to added friction, if any, appears to amount 
to about one-tenth of what is required for free maintenance. At the 
present moment it is disregarded for simplicity. 
We have then 
Hence we have 
R = - A , from r = y to T = 7r + y + 8 
= 0, elsewhere. 
with 
similarly 
hence 
ira 2 /A = ( 
• Y 
T-y+S 
cos vt dr = r 1 [sin v(tt - y 4- 8) - sin ry\ 
and therefore (p. 102) 
= 2 r 1 sin r(y - |8) cos -|-r8, 
= 2r" 1 cos r(y - J8) sin ^r8, 
-7ra 0 /A = j7r -y + J8; 
rrb r J A. = 2?’ -1 sin ?’(y - |8) sin |r8, 
= - 2 r~ l cos ?•( y - J8) cos J?-8 ; 
R = — 2A/7r[^(7r — 2y + 8) + cos (y — tj8) sin (r — ^8) 
- \ siti 2(y - J8) cos 2 (t - J8) 
+ J cos 3(y - J8) sin 3(r - J8) . 
[r even, 
[r odd. 
[r even. 
[?• odd. 
•] 
X = 2A(7 r?i 2 ) _1 [ - - 2y + 8) + (u/k) sill J8 cos (y - £8) sin r 
- J sin 2(y - J8) cos 2(r - JS) 
+ cos 3(y - J8) sin 3(r - |8) . . .] 
The first term is constant, and represents a permanent shift of the pendulum 
laterally, superposed upon the fluctuating terms. The unilateral character 
of the maintenance is noted, by a comparison of this expression with the 
expression for the Synchronome clock. This feature characterised by the 
presence of even harmonics, would disappear if we took Tiede’s escapement 
in place of Gill’s. 
Also 
n'ln= {1 + (k/??) 2 cot 2 J8}- 4- J(#c/w) cot ^8 . . . (22) 
