93 
1917-18.] Studies in Clocks and Time-keeping. 
In the same way it is seen 
( 
exp v = exp(- ~rj 1 1 - ^ ~ - ~ sin (2 n't + 2e 0 ) + T \^- 2 cos (2 n't + 2e 0 ) 
+ A C0S ( 4 ^ + 4e 0 ) . . . ) 
n - J 
Here, again, n' may be replaced by n. 
For most purposes the first approximation suffices, viz., 
where 
x = a sin r, 
a = a 0 exp ^ — \td - J — sin 2(?z£ + e Q )^ 
(126) 
(12c) 
r = lit -f € 0 — | — cos 2{nt + € 0 ), 
71 
which shows clearly the fluctuation of instantaneous amplitude and epoch 
of the vibration. 
Repeating the foregoing, but taking the second choice indicated on p. 91, 
viz. n' = n, write for this special case 
= nt + e, 
where e now contains a term proportional to the time ; and 
x = a 0 exp v . sin <f>, dxjdt = na Q exp v . cos cf> (9) 
so that 
then from (10) 
whence 
or 
where 
71 
tan = — tan r ; 
n 
dxfi/dt = n + k sin cos <p, 
ndt = d (tan <£)/ ( (l - ^ 2 ) + ( 2 1 + tan <t> 
n tan </> + tan (nt + C) 
- — n 2 — 1//2 
4 5 
(13) 
which agrees with equation (11c). 
The frequency of <p is n' ; for a first approximation, the equations (12c) 
hold when cp replaces r ; for a second approximation 
id , i K . i id . / c\ , tr\ \ i K 2 
= e 0 - i - i - cos ( 2 nt + 2<r o) - tV- 9 sin ( 2nt + 2e o) - X2 -9 sin ( int + 4e o) • ( 14 ) 
n 
n 
n- 
Before taking the case of maintained motion, in which R^O, consider 
the modification of the fundamental equation (8) if terms of second and 
third orders in x, clxjdt are present owing to geometrical or dynamical 
causes. 
