81 
1917-18.] Studies in Clocks and Time-keeping. 
so that the equation for Q is 
Q cos 0 = ra(/— V 2 /« cos 3 0) 
and contact ceases at the position 0 — y, where 
cos 3 y = Y 2 jaf (1) 
The whole impulse conveyed to the pendulum is 
but 
hence 
or 
I = J dt . Q sin 0 ; 
dt = a cos OdO/Y ; 
1 = 
maf I "cos w 1 - - sec 2 e) tan 6 = 
Y 
ajt 
Y 
a V 2 ~]y 
cos 0 sec 2 0 
2a f Jo 
1 = ^(1 - cos y) 2 (l +i cos y). 
(2) 
The work done upon the pendulum by the maintenance in 1 sec. is 
equal to the product IV. 
Hence it is evident that the impulse falls off faster than in proportion 
to any increase in V, the velocity of the pendulum, which is proportional 
to a , the full semi-arc of the swing ; using the equations 
SV/Y = Sa/a, 23Y/Y = - 3 tan y S y , 
we have 
81 ,4 COS y J cos y 
1 _ + 3(1 — COS y) " 1 + tt COS y 
The factor in [ ] expresses the efficacy of the reduction for different 
values of y ; its value is given below, as well as the ratio of I to maf/Y : — 
VOL. XXXVIII. 6 
ba 
a 
■ (3) 
