Having thus shown how the lever with its springs can be made to serve a variety of purposes, we return to the description of the actual instrument constructed. The machine serves first of all to sum up a series of harmonic motions or to draw the curve
Y = a1 cos x + a2 cos 2x + a3 cos 3x + . . . (2)
The motion of the points P1P2 ... is here made harmonic by aid of a series of excentric disks arranged so that for one revolution of the first the other disks complete 2, 3, ... revolutions. They are all driven by one handle. These disks take the place of the templets described before. The distances NG are made equal to the amplitudes a1, a2, a3, ... The drawing-board, moved forward by the turning of the handle, now receives a curve of which (2) is the equation. If all excentrics are turned through a right angle a sine-series can be added up.
It is a remarkable fact that the same machine can be used as a harmonic analyser of a given curve. Let the curve to be analysed be set off along the levers NG so that in the old notation it is
y″ = f(x),
whilst the curves y′ = φ(xξ) are replaced by the excentrics, hence ξ by the angle θ through which the first excentric is turned, so that y′k = cos kθ. But kh = x and nh = π, n being the number of springs s, and π taking the place of c. This makes
| kθ = | n π | θ.x. |
Hence our instrument draws a curve which gives the integral (1) in the form
| y = | 2 π | ∫0π f(x)cos | n π | θx | dx |
as a function of θ. But this integral becomes the coefficient am in the cosine expansion if we make