The integral can be made more general by varying the distances NG = y″. These can be set to form another curve y″ = f(x). We have now y = μy′y″ = μ f(x) φ(x), and get as before
| ∫0cf(x) φ(x)dx = | λh μ | z, |
These integrals are obtained by the addition of ordinates, and therefore by an approximate method. But the ordinates are numerous, there being 79 of them, and the results are in consequence very accurate. The displacement z of B is small, but it can be magnified by taking the reading of a point T′ on the lever AB. The actual reading is done at point T connected with T′ by a long vertical rod. At T either a scale can be placed or a drawing-board, on which a pen at T marks the displacement.
If the points G are set so that the distances NG on the different levers are proportional to the terms of a numerical series
u0 + u1 + u2 + ...
and if all P be moved through the same distance, then z will be proportional to the sum of this series up to 80 terms. We get an Addition Machine.
The use of the machine can, however, be still further extended. Let a templet with a curve y′ = φ(ξ) be set under each point P at right angles to the axis of x hence parallel to the plane of the figure. Let these templets form sections of a continuous surface, then each section parallel to the axis of x will form a curve like the old y′ = φ(x), but with a variable parameter ξ, or y′ = φ(ξ, x). For each value of ξ the displacement of T will give the integral
Y = ∫0c f(x) φ(ξx) dx = F(ξ), . . . (1)
where Y equals the displacement of T to some scale dependent on the constants of the instrument.
If the whole block of templets be now pushed under the points P and if the drawing-board be moved at the same rate, then the pen T will draw the curve Y = F(ξ). The instrument now is an integraph giving the value of a definite integral as function of a variable parameter.