The area generated is lp + ½ l2θ, or, expressing p in terms of w, lw + (½l2 - lc)θ. For a finite motion we get the area equal to the sum of the areas generated during the different steps. But the wheel will continue rolling, and give the whole roll as the sum of the rolls for the successive steps. Let then w denote the whole roll (in fig. 10), and let α denote the sum of all the small turnings θ; then the area is
P = lw + (½l2 - lc)α . . . (1)
Here α is the angle which the last position of the rod makes with the first. In all applications of the planimeter the rod is brought back to its original position. Then the angle α is either zero, or it is 2π if the rod has been once turned quite round.
Hence in the first case we have
P = lw . . . (2a)
and w gives the area as in case of a rectangle.
In the other case
P = lw + lC . . . (2b)
where C = (½l-c)2π, if the rod has once turned round. The number C will be seen to be always the same, as it depends only on the dimensions of the instrument. Hence now again the area is determined by w if C is known.
