Mathematical Instruments

P1.

II., fig. 2 shows the first complete working machine made on the lines of the above model. There are ten wheels, one for each simple harmonic constituent obtained by means of the har monic analyser, and the curve representing a year's tide for any port can be drawn in about four hours. The machine was con structed by A. Lege, under the superintendence of E. Roberts, who was also responsible for the design of later machines of larger capacity. In these machines the horizontal component was eliminated, the portions of the flexible wire between the upper and lower pulleys remaining always vertical. In the Liverpool machine made in 1924, there are 26 constituents ; in the latest American machine (constructed 1896-191o) there are 37 constitu ents, and the tidal curves for 7 years can be run off in 12 hours.

A. E. Donkin in 1873 designed and constructed a harmonic integrator for compounding two simple harmonic motions. The curves are drawn by a pen on a paper secured round the surface of a cylinder. By means of two eccentrics simple harmonic mo tions are given to the pen and the cylinder respectively, the rela tive number of vibrations being variable by means of change wheels. Since both pen and cylinder move at once, the curve drawn shows the combination of the two motions.

The machine shown in P1. II., fig. 3 was designed by Michelson and Stratton in 1898. The principle adopted is that of the addition of the elastic forces of spiral springs. In 1897 a machine of this type with 20 elements was made, and in the following year one with 8o elements, as in the example shown. An element con sists of an eccentric (near the base of the machine) which, by means of an eccentric rod, communicates a simple harmonic motion to the end of a horizontal lever, curved to a radius equal to the length of a long rod, the foot of which may be clamped in any position along the lever. The top end of this rod actuates

a lever whose end is attached to a small spring. Each of the 8o elements is similarly constructed, and the amplitude of the har monic motion transmitted to the end of each spring is proportional to the distance of the foot of the corresponding long rod from the middle of the curved lever; for setting these distances accu rately a special gauge is provided. The lower end of each of the small springs is attached to one end of a wide balance lever (made as a hollow cylinder on axial knife-edges), and the sum of their efforts is balanced by the action of a single powerful counter-spring. The motion of the lower end of the large spring is accordingly proportional to the algebraic sum of the motions of the upper ends of the small springs, and this resultant motion is magnified mechanically and conveyed to a pen, which registers the motion on a paper carried by a travelling plate driven by hand through the mechanism which rotates eccentrics.

By means of suitable toothed wheels forming a cone, the eccentrics are given periods increasing in regular succession ; the eccentric nearest the hand wheel revolving So times while that at the opposite end revolves once. Turning the hand wheel pro duces at the upper ends of the small springs motions correspond ing to cog+ , cos20, cos30, etc., up to cos8o0, with amplitudes depending on the setting of the long rods.

The motions of the elements may be changed from those for cosine to those of sine by disengaging the cone and turning all the eccentrics through 90°, for which purpose a long pinion is provided. The machine is used as an analyser for finding the coefficients in a Fourier's series for a given periodic function.