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 y1, y2, ... yn. 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
| = | ∑y
|
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, &c., be numbered C0, C1, C2 ... 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 C0, C1, C2 ... follow at equal distances, say each equal to h. The point Ck 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 ∫0cφ(x)dx.
Approximately this equals ∑hy = h∑y. Hence we have
| ∫0cφ(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.