The first integraph made commercially was invented independ ently by Abdank-Abakanowicz about the same time as that of Boys. This instrument has been made in considerable numbers, with modifications and improvements in design and construction, by Coradi of Zurich. Several polar integraphs have also been de signed by Prof. Pascal. (See his I Miei Integrafi, Naples, An integraph has been developed recently by V. Bush, F. D. Gage and H. R. Stewart which plots continuously a curve repre senting the product of two functions introduced into the machine in the form of curves. It evaluates F (x) against (x) from the ex pression F(x) = f fi(x)f2xdx, where and f2 are known functions, a formal or empirical. A full account of this machine is given by V. Bush and others in the Journal of the Franklin Institute (Jan. and Nov. 1927).
Harmonic Analysers.—In many scientific investigations the results of observations when plotted on paper take the form of an irregular curve which repeats itself at approximately regular intervals—i.e., the curve, is periodic. Such a curve may be con sidered to be the sum of a series of simple harmonic curves, and it is the first object of harmonic analysis to find these simple component curves, which together build up a given periodic curve.
The various arithmetical or graphical methods which have been devised for this purpose are somewhat laborious, and Lord Kelvin in 1876 was the first to invent an instrument for performing the operation mechanically. This instrument was an adaptation of the disc-sphere-cylinder planimeter invented by his brother, James Thomson, in 1876.
The first completed instrument designed by Kelvin, and used for the harmonic analysis of tidal observations, is shown in Pl. I., fig. 9. It embodies I I sets of the disc-sphere-cylinder combination, one for each harmonic. The curve to be analysed is wound on a central cylinder, and the simple harmonic angular motions of the proper periods are communicated to the disc by suitable gearing. The bar to which the tracer is attached has a series of pairs of projections which embrace the spheres. In actual use, the tracer is made to follow the curve, and the readings on the different inte grating cylinders give the required coefficients.
Other harmonic analysers have been invented in 1894 by Henrici and Sharp, by Yule in 1895, Michelson and Stratton in 1898, Mader in 1909 and Boucherot in 1913. An example of Henrici's instrument, made by Coradi in 1894 is shown in P1. II., fig. 1. A full description is given by Henrici in Phil. Mag. for July 1894; and in Ency. Brit., loth ed., art. "Mathematical Instruments." A different type of harmonic analyser, in which the principle of action is based on Clifford's graphic method of harmonic analysis, was invented by 0. Mader in 1909. An ordinary polar
planimeter forms part of the instrument, and the tracer can be adjusted on its arm so as to suit any length of base from 20 mm. to 360 mm. Previous harmonic analysers could only be applied to curves of a fixed base; thus curves to any other base required redrawing to the given base before being analysed. In using the instrument, the guide ruler is placed parallel to the base line of the curve to be analysed, and the tracer of the planimeter is placed in one of the two holes of a toothed disc. These discs are easily interchanged.
Tide-predicting Machine.—The method adopted by Kelvin is represented by the original model of his tide-predicting ma chine, made in 1872, and preserved in the Science Museum. In this model, eight pulleys are carried on axes at the ends of eight cranks of adjustable length, four on the upper side and four on the lower side of a rectangular wooden frame. A cord fixed at one end passes alternately under and over the lower and upper pulleys respectively, and at the other end carries a weight representing the marker. The centre of each pulley can thus describe a circle of adjustable radius, which circular motion is equivalent to the sum of two simple harmonic motions, one vertical and the other horizontal. The horizontal component of the circular motion leads to a slight motion of the cord out of its vertical position. If the radius of the circle described by the centre of each pulley is a small fraction of the distance between the upper and lower pul leys, Kelvin considered that the error introduced was practically negligible. The hanging weight will therefore perform a complex harmonic motion, which is the sum of the constituent vertical harmonic motions of the pulleys.