14,6 Mr. Atwood's Investigations for determining 
x a circular arc of which the 
sine 
b z + nc z — 2 h d z 
The result is, that when the points of quiescence of the 
auxiliary springs are situated in the latter semiarcs of the re- 
spective vibrations, the time in which the balance describes the 
x a circular arc of which 
semiarc 
B O will be = 
if x n + 
the sine is 
+ v/= 
2 If X i + 2 n 
■ x a circular arc 
y 
d z x I + 
of which the sine is ^ b * + nc *_ 2 n d z 
The balance of Mr. Mudge's time-keeper describes the se- 
miarc B O, by the joint action of two springs, i. e. the balance* 
spring, and an auxiliary spring; each spring is wound through 
the same arc B O, and comes to the same point of quiescence 
O (fig. 6.) ; consequently the action of the two springs is the 
same with that of a single spring of equal strength with both. 
In this case, the time of a semivibration through the arc B O, 
will be obtained from a former solution ; for referring to page 
126, and making O D = a, and the force of the balance spring 
and auxiliary spring at the distance from quiescence O D, == 
f jl n f— F ; the time of a semivibration is 
/ _ /JF_ 
~ V sTTTt+t “ ^ 8 / r 
But if the points of quiescence of the balance and auxiliary 
springs instead of coinciding, according to the principle of Mr. 
Mudge’s construction, should deviate from this adjustment 
by a small arc O Q = ON (fig. 4 and 5.) ; to what extent the 
* A double spiral spring is applied in the balance of Mr. Mudge’s time-keeper, 
but as these two springs act as one spring, they are here considered as such. 
