101 
1917-18.] Studies in Clocks and Time-keeping. 
simplified by integrating for mean over several periods. Denoting such 
means by [ ], we have, to the first order in k, 
[E'/E] = -k+[H/E] (19) 
[2e>[-*/E] ; 
if we proceed to the second order in k/u, the term —K 2 /4>n is introduced on 
the right-hand member of the second equation, besides possibly also terms 
depending upon products of k and the coefficients of 
To compare this with the former solution, when a steady state of 
maintenance has been reached, write 
x = /3 sin <f>. x = n/3 cos <£, E = 1 2 /3 2 , 
R = A cos <f> 4- B sin <£, 
H = RE, <I> = Hx ; 
hence in the steady state 
0 = - k + A/n/3 (20) 
2e' = -B /n 2 f3 = -/<B/wA, 
which are equivalent to (16a), (166) of p. 97, to the order of k/u. 
We may now enumerate the data which are required before the steady 
motion can be calculated. They are A, B, k, n, e 0 . Of these, in any single 
case of motion, e 0 gives merely an origin from which the measurements of 
time are made, and need not concern us further, n may be taken as known, 
viz. 7 r/n — l sec. To determine A, B, k we require to observe (1) intensity 
of the maintenance, (2) its phase with respect to oscillation when all is 
steady, and (3) the amplitude of the arc under the same circumstances. 
Of these (3) may be accurately read to about 1 sec. in the semi-oscillation 
by means of a fixed microscope which observes alternately the points of 
instantaneous rest, left and right, of a divided scale attached to the 
pendulum ; to find (1) we can make some estimate of the rate of supply 
of energy by the maintenance, but not in all cases a very good one, for 
though a design of the clock is always aimed at which measures out 
geometrically a constant quantity of work, the part that reaches the 
pendulum, which alone matters, may vary by abstractions due to its mode 
of transmission ; the phase (2), measured by B/A, is generally given with 
some accuracy, when the total arc is known, from the geometrical circum- 
stances of application of the impulse with regard to the mean. 
It will appear further on that the theory gives a fair account of the 
value a of the amplitude of oscillation, but with regard to the time- 
keeping, the position is more complicated, and must be reserved for future 
examination. 
Return now to the expression for the maintenance as a Fourier series. 
