89 
1917-18.] Studies in Clocks and Time-keeping. 
The frequency of the free oscillation is n where 
{(£? + ZA 0 )(d - IA 3 ) + k 2 }n 2 = g{d+ (A 2 - A 3 )J} . . . (7a) 
This equation shows how the bending and reaction of the spring affect the 
motion. It is usually stated that the pendulum behaves as if it turned 
about a point midway in the length of the spring ; geometrically, this is 
approximately true ; the terms on the left expressing the kinetic reactions 
are nearly equal to 
M{ (d + |Z) 2 + fc 2 }0', since A 0 -A 3 =l ; 
but the restoring force brought in by the bending couple of the spring does 
not fit in with this statement even approximately ; in place of a coefficient 
we have for the two cases considered 
d+3-lll and d + 0'591 
respectively. The change comes in chiefly through the coefficient A 2 , 
expressing the dependence of the bending couple upon the displacement 
of the pendulum. 
The correction to the term d, viz. (A 2 — A 3 )£ = A -1 { G°s ec liL A + th(A/2)}£ 
= A -1 coth A. A Now A -1 c< (bc 3 fl ~ 1 ; hence this correction a (Ac 3 )~ J coth A ; 
its value for a very weak spring suspension (A = oo ) is zero, corresponding 
to the case coth oo = 1 ; as the stoutness of the spring is increased or the 
length diminished the correction increases in value and the clock gains 
more than in proportion with the diminished length. 
From this it will be seen that the rule that the effective pivot of the 
pendulum is half-way in the length of the pendulum spring must never 
be applied as a dynamical rule but merely as a geometrical one. 
For example, the length of the “simple equivalent pendulum ” is not 
d-\-^i-\- . . . , but d-j~dAo — A 2 ) -(- . . . 
In the case of Riefler’s maintenance, P and PA are zero, and in place of 
the latter we have a given applied couple figuring in the last of the 
equations of motion (6). Hence there is no essential difference in the 
equation for Riefler, and it does not need a separate discussion. 
We shall take the equation of motion, derived from (7), in the form 
x" + . . . + n 2 x = R, 
in which x may be identified at sight with the arc 0 or some multiple of it, 
and rd is defined by the equation (7a) 
{d 2 -4- . . . + k 2 }n 2 = g{d + (A 2 — A 3 )Z}. 
The blank represents the dissipative terms, to which I now proceed. 
The effect of friction upon the pendulum, its law and intensity, requires 
to be determined experimentally. A separate study will be devoted to 
