ar to that given by F.W. Hill (_Phil. Mag._ 1894), is too complicated
to repeat here. The integrals expressing the area generated by QT have to
be expanded in a series. By retaining only the most important terms a
result is obtained which comes to this, that if the mass-centre of the area
be taken as R, then A may be any point on the curve. This is only
approximate. Captain Prytz gives the following instructions:--Take a point
R as near as you can guess to the mass-centre, put the tracer T on it, the
knife-edge Q outside; make a mark on the paper by pressing the knife-edge
into it; guide the tracer from R along a straight line to a point A on the
boundary, round the boundary, [v.04 p.0978] and back from A to R; lastly,
make again a mark with the knife-edge, and measure the distance c between
the marks; then the area is nearly cl, where l = QT. A nearer approximation
is obtained by repeating the operation after turning QT through 180 deg. from
the original position, and using the mean of the two values of c thus
obtained. The greatest dimension of the area should not exceed 1/2l,
otherwise the area must be divided into parts which are determined
separately. This condition being fulfilled, the instrument gives very
satisfactory results, especially if the figures to be measured, as in the
case of indicator diagrams, are much of the same shape, for in this case
the operator soon learns where to put the point R.

Integrators serve to evaluate a definite integral [Integral,a:b] f(x)dx. If
we plot out [Sidenote: Integrators.] the curve whose equation is y = f(x),
the integral [Integral]ydx between the proper limits represents the area of
a figure bounded by the curve, the axis of x, and the ordinates at x=a,
x=b. Hence if the curve is drawn, any planimeter may be used for finding
the value of the integral. In this sense planimeters are integrators. In
fact, a planimeter may often be used with advantage to solve problems more
complicated than the determination of a mere area, by converting the one
problem graphically into the other. We give an example:--

[Illustration: FIG. 18.]

Let the problem be to determine for the figure ABG (fig. 18), not only the
area, but also the first and second moment with regard to the axis XX. At a
distance a draw a line, C'D', parallel to XX. In the figure draw a number
of lines parallel to AB. Let CD be one of them. Draw C and D vertically
upwards to C'D', join these points to some point O in XX