a (1), [alpha] = 0; and hence

P = lw,

[v.04 p.0977] which is read off. But if the area is too large the pole O
may be placed within the area. The rod describes the area between the
boundary of the figure and the circle with radius r = OQ, whilst the rod
turns once completely round, making [alpha] = 2[pi]. The area measured by
the wheel is by formula (1), lw + (1/2l squared-lc) 2[pi].

To this the area of the circle [pi]r squared must be added, so that now

P = lw + (1/2l squared-lc)2[pi] + [pi]r squared,

or

P = lw + C,

where

C = (1/2l squared-lc)2[pi] + [pi]r squared,

is a constant, as it depends on the dimensions of the instrument alone.
This constant is given with each instrument.

[Illustration: FIG. 13.]

[Illustration: FIG. 14.]

Amsler's planimeters are made either with a rod QT of fixed length, which
gives the area therefore in terms of a fixed unit, say in square inches, or
else the rod can be moved in a sleeve to which the arm OQ is hinged (fig.
13). This makes it possible to change the unit lu, which is proportional to
l.

In the planimeters described the recording or integrating apparatus is a
smooth wheel rolling on the paper or on some other surface. Amsler has
described another recorder, viz. a wheel with a sharp edge. This will roll
on the paper but not slip. Let the rod QT carry with it an arm CD
perpendicular to it. Let there be mounted on it a wheel W, which can slip
along and turn about it. If now QT is moved parallel to itself to Q'T',
then W will roll without slipping parallel to QT, and slip along CD. This
amount of slipping will equal the perpendicular distance between QT and
Q'T', and therefore serve to measure the area swept over like the wheel in
the machine already described. The turning of the rod will also produce
slipping of the wheel, but it will be seen without difficulty that this
will cancel during a cyclical motion of the rod, provided the rod does not
perform a whole rotation.

[Illustration: FIG. 15.]

The first planimeter was made on the following principles:--A frame FF
(fig. 15) can move parallel to OX. It carries a rod TT [Sidenote: Early
forms.] movable along its own length, hence the tracer T can be guided
along any curve ATB. When the rod has been pushed back to Q'Q, the tracer
moves along the axis OX. On the frame a cone VCC' is mounted with its axis
sloping so that its top edge is horizontal and parallel to TT', whilst its
vertex V is oppos