Isochronism
A balance spring obeys Hooke's Law: the restoring torque is proportional to the angular displacement. When this property is exactly satisfied, the balance spring is said to be isochronous, and the period of oscillation is independent of the amplitude of oscillation. This is an essential property for accurate timekeeping, because no mechanical drive train can provide absolutely constant driving force. This is particularly true in watches and portable clocks which are powered by a mainspring, which provides a diminishing drive force as it unwinds. Another cause of varying driving force is friction, which varies as the lubricating oil ages.
Early watchmakers empirically found approaches to make their balance springs isochronous. For example, John Arnold in 1776 patented a helical (cylindrical) form of the balance spring, in which the ends of the spring were coiled inwards. In 1861 M. Phillips published a theoretical treatment of the problem. He demonstrated that a balance spring whose center of gravity coincides with the axis of the balance wheel is isochronous.
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