figure. Imagine now that the disks B and C also receive arms of length l
from the centres of the disks to points T_1 and T_2, 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 T_1 and
T_2 will describe closed curves, F_1 and F_2, and the "rolls" of W_1 and
W_2 will give their areas A_1 and A_2. Let XX (fig. 20) denote the line,
parallel to the rail, on which O moves; then when T lies on this line, the
arm BT_1 is perpendicular to XX, and CT_2 parallel to it. If OT is turned
through an angle [theta], clockwise, BT_1 will turn counter-clockwise
through an angle 2[theta], and CT_2 through an angle 3[theta], also
counter-clockwise. If in this position T is moved through a distance x
parallel to the axis XX, the points T_1 and T_2 will move parallel to it
through an equal distance. If now the first arm is turned through a small
angle d[theta], moved back through a distance x, and lastly turned back
through the angle d[theta], the tracer T will have described the boundary
of a small strip of area. We divide the given figure into [v.04 p.0979]
such strips. Then to every such strip will correspond a strip of equal
length x of the figures described by T_1 and T_2.

The distances of the points, T, T_1, T_2, from the axis XX may be called y,
y_1, y_2. They have the values

y = l sin [theta], y_1 = l cos 2[theta], y_2 = -l sin 3[theta],

from which

dy = l cos [theta].d[theta], dy_1 = - 2l sin 2[theta].d[theta], dy_2 = -
3l cos 3[theta].d[theta].

The areas of the three strips are respectively

dA = xdy, dA_1 = xdy_1, dA_2 = xdy_2.

Now dy_1 can be written dy_1 = - 4l sin [theta] cos [theta]d[theta] = - 4
sin [theta]dy; therefore

dA_1 = - 4 sin [theta].dA = - (4/l) ydA;

whence

A_1 = - 4/l [Integral]ydA = - 4/l A[=y],

where A is the area of the given figure, and [=y] the distance of its
mass-centre from the axis XX. But A_1 is the area of the second figure F_1,
which is proportional to the reading of W_1. Hence we may say

A[=y] = C_1w_1,

where C_1 is a constant depending on the dimensions of the instrument. The
negative sign in the expression for A_1 is got rid of by numbering the
wheel W_1 the other way round.

Again

dy_2 = - 3l cos [theta] {4 cos squared [theta] - 3} d[theta] = - 3 {4 cos squared
[theta] - 3} dy

= - 3 {(4/l squared) y squared - 3} dy,