a]; then the area is

P = lw + (1/2l^2 - lc)[alpha] . . . (1)

Here [alpha] 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 [alpha] is either zero, or it is 2[pi] 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 = (1/2l-c)2[pi], 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.

[Illustration: FIG. 11.]

Thus it is seen that the area generated by the motion of the rod can be
measured by the roll of the wheel; it remains to show how any given area
can be generated by the rod. Let the rod move in any manner but return to
its original position. Q and T then describe closed curves. Such motion may
be called cyclical. Here the theorem holds:--_If a rod QT performs a
cyclical motion, then the area generated equals the difference of the areas
enclosed by the paths of T and Q respectively._ The truth of this
proposition will be seen from a figure. In fig. 11 different positions of
the moving rod QT have been marked, and its motion can be easily followed.
It will be seen that every part of the area TT'BB' will be passed over once
and always by a _forward motion_ of the rod, whereby the wheel will
_increase_ its roll. The area AA'QQ' will also be swept over once, but with
a _backward_ roll; it must therefore be counted as negative. The area
between the curves is passed over twice, once with a forward and once with
a backward roll; it therefore counts once positive and once negative; hence
not at all. In more complicated figures it may happen that the area within
one of the curves, say TT'BB', is passed over several times, but then it
will be passed over once more in the forward direction than in the backward
one, and thus the theorem will still hold.

[Illustration: FIG. 12.]

To use Amsler's planimeter, place the pole O on the paper _outside_ the
figure to be measured. Then the area generated by QT is that of the figure,
because the point Q moves on an arc of a circle to and fro enclosing no
area. At the same time the rod comes back without making a complete
rotation. We have therefore in formul