88 
Proceedings of the Royal Society of Edinburgh. [Sess. 
Equations (5a) (56) show the couple developed and the displacement 
of any given spring of negligible mass. X= c© corresponds to the case of a 
perfectly flexible string suspension, giving L = 0, y = l(0-\-S/T) = l0, for in 
this case the shear is zero also ; as to the value of X in actual cases 
X = V(X/E), E = T \qbc*, 
where q = Young’s modulus, b — breadth and c thickness of the spring. In 
the three chief clocks considered, c is in the neighbourhood of OT mm. ; 
for Riefler and Cottingham, 6 is 0*4 cm., I = 02 cm. ; for the Synchro- 
Home clock, 6 is 1*2 cm., 1 = 1 cm. X may be taken as a weight of 7 kg., 
and q 2*0 xlO 12 dynes/cm. 2 ; the respective values of X and the other 
functions are 
R., C. 
S. 
X . 
. 0-6 
1-8 
shX 
•637 
2-942 
th(A/2) . 
•291 
•716 
A 0 
•51 
•60 
V 
•03 
•20 
A 2 
2-62 
T9 
a 3 
-•49 
-•40 
Now take the equations of motion of the pendulum itself. There are 
two cases, viz. when the maintenance P is applied by a pressure on the 
pendulum, and when, as in Riefler, it is applied by means of a couple 
impressed through the spring. 
Take the former case, and let li be the distance of the point of applica- 
tion of P below the end of the pendulum ; denoting differentiations by 
accents, 
lsl(dh n + y" - x'O) = - M g sin 0 + P — S, 
0 = — M g cos 0 + T, 
M { ( d 2 + W)V + y"d - x'd6} = - M yd sin 6 + PA L, 
or if the first power of 0 only is kept, 
M(d + l A o )0" = — sin 0 + P - S, 
0= - Mc/ + T, 
M {d 2 + k 2 + dlA Q }0" = - M yd sin 0 + PA - L . . . (6) 
Eliminating S, T, L by means of (5a), 
M{(c? + lA 0 )(d— lA g ) + k 2 )0" = -M>{rf + (A 2 -A s )Z}0 + P(ft-A 8 Z) . (7) 
Retention of sin 0, cos 0 to second order would have the effect of supple- 
menting the right-hand member by the terms + JMgd 3 {d + Z(SA 2 — A 3 )}. 
Where it is necessary to consider this, I shall amalgamate it with P. 
