Mr.Lancaster explained that the difficulty to be overcome in a good machine, is to make the emery cut the surface equally from centre to edge of the lens, so that the lens will neither lengthen nor shorten the curve during its production.

To quote his words: "This really involves the problem of the 'three bodies,' or disturbing forces so celebrated in dynamical mathematics, and it is further complicated by another quantity, the 'coefficient of attrition,' or work done by the grinding material, as well as the mischief done by capillary attraction and nodal points of superimposed curves in the path of the tool.

These complications tend to cause rings or waves of unequal wear in the surface of the glass, and ruin the defining power of the lens, which depends upon the uniformity of its curve.

As the outcome of much practical experiment, combined with mathematical research, I settled upon the ratio of speed between the sheave of the lens-tool guide and the turn-table; between whose limits the practical equalization of wear (or cut of the emery) might with the greater facility be adjusted, by means of varying the stroke and eccentricity of the tool.

As the result of these considerations in the construction of the machine, the surface of the glass 'comes up' regularly all over the lens; and the polishing only takes a few minutes' work--thus keeping the truth of surface gained by using a rigid tool." The machine in question consists of a revolving sheave or ring, with a sliding strip across its diameter; the said strip having a slot and clamping screw at one end, and a hole towards the other, through which passes the axis of the tool used in forming the lens,--the slot in the strip allowing the tool to give any stroke from 0 to 1.25 inch.