1917-18.] Studies in Clocks and Time-keeping. 91 
In these equations a, e are not arbitrary constants, but are variable para- 
meters, each containing an arbitrary constant (the former being « 0 ), while 
n is at present undefined, v, e are so defined that x satisfies the relations 
(9) as well as the equation (8) ; the former condition gives 
dv . de r. 
sm r + — cos r = 0, 
dt dt 
and the latter 
,dv 
n — cos r 
dt 
dz 
ri — sin t = (n r ' 2 — ri 2 ) sin r — kii cos t + Ra„ _1 e 
dt ’ 
lo~ v 
whence for the definition of v, e 
dv/dt 
cos r 
dejdt 
sin r 
Re-" 
n a n 
K cos T + 
ri 2 - ri 2 
sm T 
n 
( 10 ) 
It is well to make clear the meaning of the assumptions (9). They are, 
of course, modelled exactly on Lagrange’s method of instantaneous orbits 
in the planetary theory. The motion of the pendulum would be simply 
harmonic if k were zero ; owing to its possession of a definite value there 
is no harmonic motion with fixed amplitude and epoch that will continue to 
fit the circumstances, but at any instant one may be found that will at that 
instant fit the position and motion possessed by x. It will be seen that to 
retain an arbitrary value for ri merely throws into e the progressive part 
of t which is not covered by n’t ; there are then two obvious choices for 
n , which will be considered below : 
(1) n = mean frequency of t, so that e contains nothing but periodic 
terms ; 
(2) n —n, leaving a term in e proportional to the time. 
We shall now connect the solution proposed with the familiar solution 
of equation (1) for the case R = 0, viz. 
x = A exp ( — \ id) sin (pt + y). [p 2 = n 2 — \k 2 . 
Eliminate e by the equation 
de/dt = dr/dt — ri ; 
then if R = 0, the integral equation for t is 
n , k / -i k 
— tan t = 1 „ 
n 2 n \ in 2 . 
tan 
n 
P- 4 4)WC); 
therefore if n is the frequency of 
ri 2 = ri 2 - Ik 2 
Eliminate t from the equation for dv/dt ; integrating 
. (11a) 
