99 
1917-18.] Studies in Clocks and Time-keeping. 
Integrating, and making Ae = 0 for t = 0, 
Ae 
2 A 
= tAn' + 
B 
A 
AB 
X' 
AA 
X 
AA Ak 
X + 7 
B 
+ A 
'AA 
X' 
A K 
(1 _ Q-^K + Ak)^ 
Ak 
(1 _ e -i(*+A/c )«) 
• (176) 
Apart from the change tAn', which expresses the effect of the permanent 
change of rate reckoned from the alteration of the circumstances, the 
epoch is shifted by an angle 
B AA Ak 
aLX ” x 
1> A a 
A a 
. (17c) 
in which it will be noticed that k as a factor, and AB, do not appear. 
For Aa positive, the change is a gain or loss according as B/A is positive 
or negative. 
It may be taken that the proportional variations AA/A, etc., do not 
exceed the order of 1 per cent., and B/A will usually be less than unity, 
so that this shift may be of the order of 1°, or, say, s, 005 in time. 
The foregoing discussion is suited for the cases in which the main- 
tenance is given as an applied force : but in the case of the clock Riefler, 
where it is supplied by bending the pendulum spring, a treatment on the 
basis of energy is clearer. I shall therefore repeat the discussion, taking 
the energy of the motion as a variable. The results are general, and apply 
to any clock and not exclusively to Riefler. 
Write 
E = £{x' 2 + n 2 x 2 }, F = nxx', G = ^{x' 2 — k¥}, 
so that 
E 2 eeF 2 + G 2 ; 
E measuring the sum of variable parts of the kinetic and potential 
energies as far as the latter is due to the frictionless elastic reactions of 
the system. 
Then if 
x -f- kx -f* n 2 x = R, 
we have, — accents denoting differentiations — 
E' = x x" + n 2 xx = — kx 2 + Vx = — k(E -1- G) + RF, 
F' = nxx" + nx 2 = — iikxx + n(x 2 — n 2 x 2 ) + 7iVx — — kF + 2nG + nJix, 
G' - xx — n 2 xx — E' — 2?iF, 
which may be written 
(E' + kE) + kG = RF =H, 
(F' + kF) — 2nG = 7iRx = 
kE + 2nF + (G' + kG) = RF - H ; 
. (18a) 
