The following problem presented itself to me nearly thirty years ago. I cannot find any notice of it in books, though it must have occurred to every one who has studied the oscillations of a balance:—

Two equal masses are attached to the ends of a cord passing over a smooth pulley (as in Attwood's machine). One of them is slightly disturbed, in a vertical plane, from its position of equilibrium. Find the nature of the subsequent motion of the system.

The interest of this case of small motions is twofold. From the peculiar form of the equations of motion, it is of exceptional mathematical difficulty. This is probably the reason for its not having been given as an example in Kinetics. And from the physical point of view it presents a very beautiful example of excessively slow, but continued, transformation of mixed potential and kinetic energy into kinetic energy alone.