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.