113 
1917-18.] Studies in Clocks and Time-keeping. 
The agreement is rough, compared with that for the other two clocks, 
but the data are not so well determined. In particular, k/u, which was 
determined from a single occasion when an accident stopped the clock, I 
should have expected to approach more nearly to the value for Cottingham, 
0‘50 x 10~ 4 , a difference that would nearly reconcile the calculation with 
observation. On the whole, I conclude from these comparisons that the 
theory developed in the foregoing pages shows itself sufficient to explain 
the arc described by the pendulum. 
Coming now to the question of time-keeping, the actual frequency is n, 
where n is given by (166) of p. 97, which for brevity I shall write in 
the form 
n'ln=\+ . . . - T>k/?i ..... ( 166 ) 
The form of D has been found for each escapement. In general D is a 
function of a, the semi-arc of complete oscillation, because, as a rule, the 
maintenance is applied at an amplitude of arc geometrically fixed, and this 
will be a different portion of the whole when the whole varies. The geo- 
metrical equations showing the relation are peculiar to each clock ; for 
example, for the Synchronome clock we have D = |tan^<5, and with 
sufficient approximation la sin S = a sin y, where cos 3 y = / 2 a% 2 /a/; for the 
others the equations may be written down as required. In conjunction 
then with equation (166), in order to find the variations of n, we must 
consider the sources of variation of a and also of n ; these are given by 
the equations 
~M(d 2 + . . . )n?a? = W(n/K) .... ( 26 ) 
A 
and 
(d~ + . . . = g(d + . . . ) j . . . .( / ciy 
further, both n 2 and k must be regarded as variable with a in accordance 
with equation (86), p. 94, 
0 = x" + [k + ^A. 2 a 2 + |A. 4 7i 2 a 2 ]af + [n- + f AjO . 2 + ^A 3 7i 2 a 2 ]it' 2 . ... . (86) 
which includes circular error and other corrections of like type. Hence we 
have the relations between the changes of these quantities 
A n 
n 
— {iV 2 +iV» 2 “ a } 
Aa 
a 
A n 
n 
AW n (k\ 
5 <Z +2 1T 2 k W 
A d Aa f 
hr + Tl 
k dY) o x 
— a— 7 - + % Aja 
n da 
2 + iV 2 “-} 
These are the equations governing the variation of frequency to which 
the foregoing theory leads. The term in An' dependent upon Ad may be 
VOL. xxxviii. 8 
