That of the lathe-band is one of those simple-looking problems in elementary mechanics which present serious difficulties to the investigator. The problem is this—“So to arrange the several diameters on the fly-wheel and pulley cone of the turning-lathe, as that the same band may suit for all.” The solution consists of two steps converse to each other, the one being “to compute the length of the band from given diameters,” the other “to compute new diameters which may suit the length so found.”