87 
1917-18.] Studies in Clocks and Time-keeping. 
Integrating the third equation 
L + Y(x- e> - X(y - rj) = Achfv/(X/E) + Bsh V(X/E), 
whence 
Xd v /d£ — Y = V(X/E) ( Ash^/(X/E) + Bch,V(X/E) }. 
To determine the constants A, B we have 
W rite 
then 
for £=0; rf — 0, drj/d^=0, 
for £=Z; y = y, dr)fd£=6. 
V(X/E) = A; 
L + Yx — Xy = A, -Yl = BA, 
L = AchA + BshA, 1X0 - IY = A{ Ash A + BchA} ; 
eliminating A, B, two equations result which give respectively y and L In 
terms of l , X, X, Y. 
From the last three equations, the equation for L is 
0 = 
L, 
1X0 - IY , 
-YZ, 
chA, shA 
AshA, AchA 
multiply the first row by XshX, the second by — chX, and add to the 
third ; then 
0 = LAshA - (1X0 - 1 Y)chA - YZ, 
or 
AL = 1X0/ coth A - ZYth(A/2), 
= ITO/shX — ZSth(A/2), 
or say 
L = T . 10 . A 2 + S . I . A 3 , (5a) 
where 
A 2 = 1/AshA, A 3 = — A _1 th(A/2). 
Similarly from the first three equations 
0 = 
L + Yx — Xy, 
- 1 , 
* 
— L, chA, shA 
Y Z, *, A 
or 
(L + Yx — Xy) AchA — YZshA — LA = 0. 
Substituting for L, and putting x = l, we find 
y = 10 . A _1 th(A/2) + Z(Y/X){1 - 2A-Hh(A/2)} 
= 10{\ - A-Hh(A/2)} + Z(S/T){1 - 2A“ 1 th(A/2)} 
= Z^A t) + Z(S/T)A 1 (56) 
say, where 
— =5 <n.J Aq =1 — A-Hh(A/2), A 1= = 1 - 2A-Hh(A/2), 
so that 
A-Hh(A/2) = 1 - A 0 = i(l -Aj)= -A 3 . 
