97 
1917-18.] Studies in Clocks and Time-keeping. 
part of a has been effaced. This removes also the arbitrary constants of 
integration of the differential equation, corresponding to the arbitrary 
character of the way in which the motion was started. We have merely 
to find a particular integral, and this can be done by inspection ; for if we 
have the equations 
x" + kx + n 2 x = A cos r + B sin r 
where 
a solution is 
x = a sin r, 
where 
i.e. 
or 
a = a 0 = A /ku\ T = n't + e () 
ri 2 = ri 1 - B/a 0 , 
A(?7 2 — n 2 ) = — B ku 
n \ + A' 2 n 2 ) 2 An 
. (16a) 
. (166) 
These equations are exact. 
The same results are visible also as a solution of (10), p. 91, where v = 0. 
e = e 0 is seen to be a solution provided a 0 , n satisfy (16a), (166). 
The equations just given are of fundamental importance. I shall 
therefore make a few remarks upon them. 
In ordinary cases of maintained motion or forced vibration, the period 
of the maintenance is given and the forced vibration is identical with it 
and therefore given also, a definite lag separating the phases. Here, how- 
ever, the period of the maintenance is dependent upon the elastic reactions 
of system. In consequence, the phase of the maintenance as measured by 
B/A appears in the period, and with it, what is most important, the first 
power of the frictional coefficient k. It has been stated, I think without 
exception hitherto, that the friction of a vibrating system affects the 
frequency only in the second order, referring to the equation n 2 = n 2 — \k 2 > 
but this is only true of the natural period of non-maintained motion, and, 
as it appears, is quite inapplicable to the case of maintained motion, first, 
because this reaches no steady period at all until the free motion has died 
out, and, second, because the frequency to which it then settles down is 
given by (166). 
Consider one or two cases of given phase. 
(1) Suppose B = 0, so that the maintenance is symmetrical with respect 
to the zero of x and has its maximum at this zero ; then n = n, the frictional 
coefficient k does not enter the period, and the phase of the maintained 
motion, x, is always a quarter period from that of the maintenance. 
(2) Suppose A = 0, or the maintenance has its maximum at the extreme 
VOL. xxxviii. 7 
