138 Mr. Atwood's Investigations for determining 
the same angle O C D = Q C d, in the proportion of 1 to n ; 
then the accelerative force of the auxiliary spring at the an- 
gular distance from quiescence O d will = nf ; let B O = b , 
B Q = c, QO = d = b — c ; also let O D or Q d = a. 
Suppose the balance to have described the arc B H by the 
joint action of both the springs, and let the arc B H be repre- 
sented by x. Then, because the accelerative force of the prin- 
cipal or balance spring at the angular distance from quiescence 
OD = a is/, the accelerative force of the same spring at the 
distance O H is — and since the force of the auxiliary 
spring at the angular distance from quiescence Q d = a is nf, 
the accelerative force of this spring at the angular distance 
from quiescence Q H will be = wherefore the joint 
force of both springs to accelerate the circumference when at 
the distance O H from the point of quiescence O, that is, when 
the balance has described the arc B H, will be = —■ x 
jgg_ x -j- cn — n x. Let u be the space through which a body falls 
freely from rest by the acceleration of gravity, to acquire the 
velocity of the circumference when it has described the arc 
B H ; this will give the following equation : 
u — ~ — x b x — x x -j- c n x — n x x , and 
« = i.x 2 b x — x* 4- 2 cn x — n x' ; and if / = 193 inches, 
2 a 
the velocity of the circumference when the arc B H has 
been described, is — x\/ 26a: — x % + 2 c n x n x . 
Let t represent the time in which the balance describes the arc 
B H ; then i = 
X 
\/ 2 b x — x x + z c nx — nx L 
and the fluent 
or t 
If X n + 1 
x into a circular arc of which the sine is 
