Page:EB1911 - Volume 04.djvu/1005

 and back from A to R; lastly, make again a mark with the knife-edge, and measure the distance c between the marks; then the area is nearly cl, where l = QT. A nearer approximation is obtained by repeating the operation after turning QT through 180° from the original position, and using the mean of the two values of c thus obtained. The greatest dimension of the area should not exceed l, otherwise the area must be divided into parts which are determined separately. This condition being fulfilled, the instrument gives very satisfactory results, especially if the figures to be measured, as in the case of indicator diagrams, are much of the same shape, for in this case the operator soon learns where to put the point R.

Integrators serve to evaluate a definite integral $$\textstyle\int_{a}^{b}$$ ƒ(x)dx. If we plot out the curve whose equation is y = ƒ(x), the integral ∫ydx between the proper limits represents the area of a figure bounded by the curve, the axis of x, and the ordinates atx=a, x=b. Hence if the curve is drawn, any planimeter may be used for finding the value of the integral. In this sense planimeters are integrators. In fact, a planimeter may often be used with advantage to solve problems more complicated than the determination of a mere area, by converting the one problem graphically into the other. We give an example:—

18.

Let the problem be to determine for the figure ABG (fig. 18), not only the area, but also the first and second moment with regard to the axis XX. At a distance a draw a line, C′D′, parallel to XX. In the figure draw a number of lines parallel to AB. Let CD be one of them. Draw C and D vertically upwards to C′D′, join these points to some point O in XX, and mark the points C1D1 where OC′ and OD′ cut CD. Do this for a sufficient number of lines, and join the points C1D1 thus obtained. This gives a new curve, which may be called the first derived curve. By the same process get a new curve from this, the second derived curve. By aid of a planimeter determine the areas P, P1, P2, of these three curves. Then, if $\overline{x}$ is the distance of the mass-centre of the given area from XX; $\overline{x}$1 the same quantity for the first derived figure, and I = Ak2 the moment of inertia of the first figure, k its radius of gyration, with regard to XX as axis, the following relations are easily proved:—

P$\overline{x}$＝aP1; P1$\overline{x}$1＝aP2; I＝aP1$\overline{x}$1＝a2P1P2; k2＝$\overline{xx}$1,

which determine P, $\overline{x}$ and I or k. Amsler has constructed an integrator which serves to determine these quantities by guiding a tracer once round the boundary of the given figure (see below). Again, it may be required to find the value of an integral ∫y(x)dx between given limits where (x) is a simple function like sin nx, and where y is given as the ordinate of a curve. The harmonic analysers described below are examples of instruments for evaluating such integrals.

19.

20.

Amsler has modified his planimeter in such a manner that instead of the area it gives the first or second moment of a figure about an axis in its plane. An instrument giving all three quantities simultaneously is known as Amsler’s integrator or moment-planimeter. It has one tracer, but three recording wheels. It is mounted on a carriage which runs on a straight rail (fig. 19). This carries a horizontal disk A, movable about a vertical axis Q. Slightly more than half the circumference is circular with radius 2a, the other part with radius 3a. Against these gear two disks, B and C, with radii a; their axes are fixed in the carriage. From the disk A extends to the left a rod OT of length l, on which a recording wheel W is mounted. The disks B and C have also recording wheels, W1 and W2, the axis of W1 being perpendicular, that of W2 parallel to OT. If now T is guided round a figure F, O will move to and fro in a straight line. This part is therefore a simple planimeter, in which the one end of the arm moves in a straight line instead of in a circular arc. Consequently, the “roll” of W will record the area of the figure. Imagine now that the disks B and C also receive arms of length l from the centres of the disks to points T1 and T2, and in the direction of the axes of the wheels. Then these arms with their wheels will again be planimeters. As T is guided round the given figure F, these points T1 and T2 will describe closed curves, F1 and F2, and the “rolls” of W1 and W2 will give their areas A1 and A2. Let XX (fig. 20) denote the line, parallel to the rail, on which O moves; then when T lies on this line, the arm BT1 is perpendicular to XX, and CT2 parallel to it. If OT is turned through an angle, clockwise, BT1 will turn counter-clockwise through an angle 2, and CT2 through an angle 3, also counter-clockwise. If in this position T is moved through a distance x parallel to the axis XX, the points T1 and T2 will move parallel to it through an equal distance. If now the first arm is turned through a small angle d, moved back through a distance x, and lastly turned back through the angle d, the tracer T will have described the boundary of a small strip of area. We divide the given figure into