INSTRUMENTS.) The first mechanical harmonic analyzer was proposed by Lord Kelvin, who made use of an integrator invented by his brother, J. Thomson. The integrator consists of a disc D, a sphere S, and a cylinder C, as shown in fig. 4. The sphere rolls along a diameter of the disc so that when it is in the centre of D it will be unaffected by the rotation of D. At any other point it will turn an amount proportional to its distance from the centre of D.
which, by (2), is proportional to the kth sine or cosine coefficient. By having a number of such units side by side several coefficients can be obtained at once. The spheres are all moved together an amount equal to the ordinates of the curve, while the discs are turned through angles which are exact multiples of each other.
Another machine working on a similar principle is that invented by Henrici and developed by Coradi. This very beautiful instru ment makes use of a number of glass spheres which are moved an amount equal to the ordinate of the curve. The readings are obtained by two Amsler integrating wheels bearing on each sphere, one reading sine components, and the other, cosine components. A five-sphere machine will thus give ten coefficients for one trace of the curve. The machine is speedy and accurate, but its expense and delicacy have limited its use to some extent. The Michelson and Stratton analyzer will handle up to eighty coefficients at once. It works equally well as a synthesizer and has been extensively used in tide prediction. The Chubb analyzer differs from all others in that it uses a polar graph instead of a rectangular one. The curve, in the form of a cardboard template, is fastened to the table of the machine. The table moves back and forth with a sinusoidal motion as it turns, and the result is integrated by an ordinary planimeter. The machine can evaluate only one coeffi cient at a time and is therefore slow. It is very rugged, and finds considerable use in connection with polar oscillograms.
A product integraph, recently developed at the Massachusetts Institute of Technology, obtains the integral of the product of any two curves. Its primary use is in the mechanical solution of differential equations, but obviously it can be used for har monic analysis. In fact, it will develop curves in terms of various other series as well as the harmonic series of Fourier. Many other machines have been proposed, notably by Mader, Rowe, Wiechert and Sommerfeld, Bashforth, Yule, Le Conte, Terada, Dellen baugh and Woodbury. Most of these machines work on the principle previously mentioned, though the Dellenbaugh uses the schedule method, while the Woodbury is based on the Fischer Hinnen method.
A large number of methods are available for the harmonic analysis of complex waves. The direct mathematical method is unnecessarily long and arduous. This has been simplified by schedules which are fairly quick and are the methods generally used if only a small amount of such work is to be done. If much harmonic analysis is required, it is advisable to use a machine analyzer, such as the Henrici-Coradi, the Michelson and Stratton, or the Chubb. Of these the first appears to offer advantages, if five or ten coefficients are sufficient. If a very large number of terms is required, the Michelson and Stratton is advisable.
The reverse process of synthesis is also used to a considerable extent, particularly in tide prediction. Various machines have been built for this purpose, the first being designed by Kelvin. An extremely compact form of Kelvin synthesizer was built by D. C. Miller for use in his work on the theory of musical sounds. The Michelson and Stratton machine is also very good for synthesis. In fact, any of the analyzers can be used for this purpose, though some are much more convenient than others.