The absolute term A0 equals the mean ordinate of the curve, and can therefore be determined by any planimeter. The other co-efficients are
| An = | 1 π | ∫02π y cos nθ.dθ; Bn = | 1 π | ∫02π y sin nθ.dθ. |
A harmonic analyser is an instrument which determines these integrals, and is therefore an integrator. The first instrument of this kind is due to Lord Kelvin (Proc. Roy Soc., vol xxiv., 1876). Since then several others have been invented (see Dyck's Catalogue; Henrici, Phil. Mag., July 1894; Phys. Soc., 9th March; Sharp, Phil. Mag., July 1894; Phys. Soc., 13th April). In Lord Kelvin's instrument the curve to be analysed is drawn on a cylinder whose circumference equals the period c, and the sine and cosine terms of the integral are introduced by aid of simple harmonic motion. Sommerfeld and Wiechert, of Konigsberg, avoid this motion by turning the cylinder about an axis perpendicular to that of the cylinder. Both these machines are large, and practically fixtures in the room where they are used. The first has done good work in the Meteorological Office in London in the analysis of meteorological curves. Quite different and simpler constructions can be used, if the integrals determining An and Bn be integrated by parts. This gives
| nAn = - | 1 π | ∫02π sin nθ.dy; nBn = | 1 π | ∫02π cos nθ.dy. |
An analyser presently to be described, based on these forms, has been constructed by Coradi in Zurich (1894). Lastly, a most powerful analyser has been invented by Michelson and Stratton (U.S.A.) (Phil Mag., 1898), which will also be described.
The Henrici-Coradi analyser has to add up the values of dy.sin nθ and dy.cos nθ. But these are the components of dy in two directions perpendicular to each other, of which one makes an angle nθ with the axis of x or of θ. This decomposition can be performed by Amsler's registering wheels. Let two of these be mounted, perpendicular to each other, in one horizontal frame which can be turned about a vertical axis, the wheels resting on the paper on which the curve is drawn. When the tracer is placed on the curve at the point θ = 0 the one axis is parallel to the axis of θ. As the tracer follows the curve the frame is made to turn through an angle nθ. At the same time the frame moves with the tracer in the direction of y. For a small motion the two wheels will then register just the components required, and during the continued motion of the tracer along the curve the wheels will add these components, and thus give the values of nAn and nBn. The factors 1/π and -1/π are taken account of in the graduation of the wheels. The readings have then to be divided by n to give the coefficients required. Coradi's realization of this idea will be understood from fig. 23. The frame PP′ of the instrument rests on three rollers E, E′, and D. The first two drive an axis with a disk C on it. It is placed parallel to the axis of x of the curve. The tracer is attached to a carriage WW which runs on the rail P. As it follows the curve this carriage moves through a distance x whilst the whole instrument runs forward through a distance y. The wheel C turns through an angle proportional, during each small motion, to dy. On it rests a glass sphere which will therefore also turn about its horizontal axis proportionally, to dy. The registering frame is suspended by aid of a spindle S, having a disk H. It is turned by aid of a wire connected with the carriage WW, and turns n times round as the tracer describes the whole length of the curve. The registering wheels R, R′ rest against the glass sphere and give the values nAn and nBn. The value of n can be altered by changing the disk H into one of different diameter. It is also possible to mount on the same frame a number of spindles with registering wheels and glass spheres, each of the latter resting on a separate disk C. As many as five have been introduced. One guiding of the tracer over the curve gives then at once the ten coefficients An and Bn for n = 1 to 5.
All the calculating machines and integrators considered so far have been kinematic. We have now to describe a most remarkable instrument based on the equilibrium of a rigid body under the action of springs. The body itself for rigidity's sake is made a hollow
Michelson and Stratton analyzer cylinder H, shown in fig. 24 in end view. It can turn about its axis, being supported on knife-edges O. To it springs are attached at the prolongation of a horizontal diameter; to the left a series of n small springs s, all alike, side by side at equal intervals at a distance a from the axis of the knife-edges; to the right a single spring S at distance b. These springs are supposed to follow Hooke's law. If the elongation beyond the natural length of a spring is λ, the force asserted by it is p = kλ. Let for the position of equilibrium l, L be respectively the elongation of a small and the large spring, k, K their constants, then
