In an Appendix to Part I an account is given of Continuous Calculating Machines. Ordinary calculating machines, such as the "arithmometer" of Thomas of Colmar, carry out calculations and exhibit the result as a row of figures. But the machines here described are of a different character: they exhibit their results by values of a continuously varying quantity. The first is one for predicting the height of the tides for future time, at any port for which data have been already obtained regarding tidal heights, by means of a self-registering tide-gauge. Two of these were made according to the ideas set forth in this Appendix; one is in the South Kensington Museum, the other is at the National Physical Laboratory at Bushy House, where it is used mainly for drawing on paper curves of future tidal heights, for ports in the Indian Ocean. From these curves tide-tables are compiled, and issued for the use of mariners and others. Another machine described in this Appendix was designed for the mechanical solution of simultaneous linear equations. It is impossible to explain here the interesting arrangement of six frames, carrying as many pulleys, adjustable along slides (for the solution of equations involving six unknown quantities), which Thomson constructed, and which is now in the Natural Philosophy Department at Glasgow. The idea of arranging the first practical machine for this number of variables, was that it might be used for the calculation of the corrections on values already found for the six elements of a comet or asteroid. The machine was made, but some mechanical difficulties arose in applying it, and the experiments with it were not at the time persevered with. Very possibly, however, it may yet be brought into use. [Illustration: FIG. 14.]