112 Proceedings of the Royal Society of Edinburgh. [Sess. 
Making the calculation with this value of y : 
Semi-arc. 
Calc. 
Obsd. 
7- 
inf = TO gr. . 
. 120' 
110' 
[57°-3] 
0-85 „ . 
. 110' 
105' 
[58°-4] 
0-55 „ . 
89' 
90' 
[60°-7] 
If we work from the observed values of a, and derive the values of y, 
the formula gives for y the angles shown enclosed in [ ]. 
The value of k/u changes by some units per cent, according to the state 
of the oil on the crutch and its link to the pendulum. 
Examining the formula of p. 103 and the coefficient of sin r in the 
expression for x, it is seen that the same numbers are derived as from the 
energy equation that has been employed above. 
Riefler . — The energy conferred in bending a straight spring of length l, 
breadth b, and thickness c, from straightness to a uniform curvature 1/p, is 
w=^# 3 /(i-A 2 
where q is Young’s modulus, and p a constant, being the ratio of lateral 
contraction to longitudinal expansion.* 
We may take approximately 
q = 2 x 10 12 , 
/* = !> 
l = Q"2 cm., 
b = 0'4 cm., 
c = 0'012 cm. 
As regards p, following the data of p. 78, the bending of the spring 
begins when the lift face of the scape-wheel comes into contact with the 
pallet of the anchor, that is, for arc of the pendulum 0 = 30' and is completed 
for 0 = 50'. The bending contributed as added energy by the maintenance 
therefore lies between 30' and 50', the rest of the bending being supplied 
from the motion of the pendulum ; taking it at 40', we have 
hence 
Also 
so that the formula 
gives 
p =10-2 x 3438'/40'= 17*2 cm., 
W = 4T2 ergs. 
M = 7 kg. 
K /n=- 77xlCr 4 ; , 
a 2 =(2/7f?l 2 )(?l/K)W/M{d 2 + . . . } 
a = 80\ 
The observed semi-arc is 95" to 100'. 
* Gf. Rayleigh, Theory of Sound, vol. i, p. 294. 
