152 Mr. Atwood's Investigations for determining 
main spring, or by removing a part of the force which retards 
the ascent of the balance while it describes the latter semiarc 
O E. Mr. Mudge has discovered and applied to his time- 
keeper the latter mode of supplying the power which is required 
to continue the balance in motion. During the progress of 
the balance through the semiarc B O, it is accelerated y the 
joint action of the balance spring and auxiliary spring u ; but 
while it describes the latter semiarc O E (fig. 7 and 8.), it is 
retarded by the joint actions of the balance spring and auxiliary 
spring, only while it describes a part of the semiarc from k to 
E, the retardation of the auxiliary spring being removed while 
it describes the first 27 0 of this semiarc from O to k. 
It is evident, according to Mr. Mudge’s construction, that 
the diminution of retardation, which is equivalent to the supply 
of power to the balance, must always be of the same magni- 
tude, so far as regards the influence of the main spring, pro- 
vided it has sufficient force to wind up the auxiliary springs 
through the constant angle OGI) = OCf (fig. 8.) ; and the 
effects of heat and cold on the auxiliary springs are necessarily 
included in the compensation which is applied for heat and 
cold to the balance spring. The maintaining power therefore, 
by which the motion of the balance is continued, must be al- 
ways uniformly the same ; this is an object usually held to be of 
material consequence in the construction of watches, and though 
often attempted by ingenious persons, has probably been ac- 
complished in its full extent for the first time by Mr. Mudge. 
This construction possesses the further advantage of having 
the balance perfectly detached from the wheel work of the ma- 
chine ; the only communication* between the balance and the 
* The exact times in which the balance describes any portions of the arc of vibra- 
tion may be readily obtained by having recourse to the theorem investigated in page 
