Problem of the isochronous hairspring.

Abstract

The system of the hairspring and balance wheel used to regulate watches and chronometers does not perform an exact simple harmonic motion. For the proper function of the watch the deviation from isochronism of the vibrations, caused by the reaction force at the fixed end point for any distortion of the spring, must be as small as possible. This reaction force is found to be correspondingly small if the shape of the hairspring is designed so that the reaction force at the end point is small.

In theory only, perfect isochronism can be accomplished by a flat spring having one end free so that for any distortion of the sprang, the tangent to the end point moves parallel to itself; however, the free end spring cannot be realised in a watch.

Phillips' solution to the problem primarily expresses that the reaction force at the end point is proportional to the displacement of the end point that would occur for a free spring. His solution is based on the problem of finding a possible curve such that the terminal point has a given trajectory or displacement, and simultaneously, at any point of the trajectory the tangent has an imposed direction. [TRUNCATED]