One guiding of the tracer over the curve gives then at once the ten coefficients A and B 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 Fig. 24.

nkla = KLb.

The position now obtained will be called the normal one. Now let the top ends C of the small springs be raised through distances y, y, ... y. Then the body H will turn; B will move down through a distance z and A up through a distance (a/b)z. The new forces thus introduced will be in equilibrium if

 ak ∑y - n a b z = bKz.

Or

z =
∑y n a b + b a K k
=
∑y n a b + l L .

This shows that the displacement z of B is proportional to the sum of the displacements y of the tops of the small springs. The arrangement can therefore be used for the addition of a number of displacements. The instrument made has eighty small springs, and the authors state that from the experience gained there is no impossibility of increasing their number even to a thousand. The displacement z, which necessarily must be small, can be enlarged by aid of a lever OT′. To regulate the displacements y of the points C (fig. 24) each spring is attached to a lever EC, fulcrum E. To this again a long rod FG is fixed by aid of a joint at F. The lower end of this rod rests on another lever GP, fulcrum N, at a changeable distance y′ = NG from N. The elongation y of any spring s can thus be produced by a motion of P. If P be raised through a distance y′, then the displacement y of C will be proportional to y′y′; it is, say, equal to μy′y′ where μ is the same for all springs. Now let the points C, and with it the springs s, the levers, etc., be numbered C, C, C ... There will be a zero-position for the points P all in a straight horizontal line. When in this position the points C will also be in a line, and this we take as axis of x.

On it the points C, C, C ... follow at equal distances, say each equal to h. The point C lies at the distance kh which gives the x of this point. Suppose now that the rods FG are all set at unit distance NG from N, and that the points P be raised so as to form points in a continuous curve y′ = φ(x), then the points C will lie in a curve y = μφ(x). The area of this curve is

μ ∫cφ(x)dx.

Approximately this equals ∑hy = h∑y. Hence we have

 ∫cφ(x)dx = h μ ∑y = λh μ z,

where z is the displacement of the point B which can be measured. The curve y′ = φ(x) may be supposed cut out as a templet. By putting this under the points P the area of the curve is thus determined - the instrument is a simple integrator.

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

 ∫cf(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

u + u + u + ...

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 = ∫c 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.

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