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 Amsler's Integrator. 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 θ, clockwise, BT1 will turn counter-clockwise through an angle 2θ, and CT2 through an angle 3θ, 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 dθ, moved back through a distance x, and lastly turned back through the angle dθ, the tracer T will have described the boundary of a small strip of area. We divide the given figure into
such strips. Then to every such strip will correspond a strip of equal length x of the figures described by T1 and T2.
The distances of the points, T, T1, T2, from the axis XX may be called y, y1, y2. They have the values
y = l sin θ, y1 = l cos 2θ, y2 = -l sin 3θ,
from which
dy = l cos θ.dθ, dy1 = - 2l sin 2θ.dθ, dy2 = - 3l cos 3θ.dθ.
The areas of the three strips are respectively
dA = xdy, dA1 = xdy1, dA2 = xdy2.
Now dy1 can be written dy1 = - 4l sin θ cos θdθ = - 4 sin θdy; therefore