ite Q'. As the frame moves it turns the cone. A wheel W is
mounted on the rod at T', or on an axis parallel to and rigidly connected
with it. This wheel rests on the top edge of the cone. If now the tracer T,
when pulled out through a distance y above Q, be moved parallel to OX
through a distance dx, the frame moves through an equal distance, and the
cone turns through an angle d[theta] proportional to dx. The wheel W rolls
on the cone to an amount again proportional to dx, and also proportional to
y, its distance from V. Hence the roll of the wheel is proportional to the
area ydx described by the rod QT. As T is moved from A to B along the curve
the roll of the wheel will therefore be proportional to the area AA'B'B. If
the curve is closed, and the tracer moved round it, the roll will measure
the area independent of the position of the axis OX, as will be seen by
drawing a figure. The cone may with advantage be replaced by a horizontal
disk, with its centre at V; this allows of y being negative. It may be
noticed at once that the roll of the wheel gives at every moment the area
A'ATQ. It will therefore allow of registering a set of values of
[Integral,a:x] ydx for any values of x, and thus of tabulating the values
of any indefinite integral. In this it differs from Amsler's planimeter.
Planimeters of this type were first invented in 1814 by the Bavarian
engineer Hermann, who, however, published nothing. They were reinvented by
Prof. Tito Gonnella of Florence in 1824, and by the Swiss engineer
Oppikofer, and improved by Ernst in Paris, the astronomer Hansen in Gotha,
and others (see Henrici, _British Association Report_, 1894). But all were
driven out of the field by Amsler's simpler planimeter.

[Illustration: FIG. 16.]

[Illustration: FIG. 17.]

Altogether different from the planimeters described is the hatchet
planimeter, invented by Captain Prytz, a Dane, and made by Herr [Sidenote:
Hatchet planimeters.] Cornelius Knudson in Copenhagen. It consists of a
single rigid piece like fig. 16. The one end T is the tracer, the other Q
has a sharp hatchet-like edge. If this is placed with QT on the paper and T
is moved along any curve, Q will follow, describing a "curve of pursuit."
In consequence of the sharp edge, Q can only move in the direction of QT,
but the whole can turn about Q. Any small step forward can therefore be
considered as made up of a motion along QT, together with a turning about
Q. The latter motion alone