, and mark the
points C_1D_1 where OC' and OD' cut CD. Do this for a sufficient number of
lines, and join the points C_1D_1 thus obtained. This gives a new curve,
which may be called the first derived curve. By the same process get a new
curve from this, the second derived curve. By aid of a planimeter determine
the areas P, P_1, P_2, of these three curves. Then, if [=x] is the distance
of the mass-centre of the given area from XX; [=x]_1 the same quantity for
the first derived figure, and I = Ak squared the moment of inertia of the first
figure, k its radius of gyration, with regard to XX as axis, the following
relations are easily proved:--

P[=x] = aP_1; P_1[=x]_1 = aP_2; I = aP_1[=x]_1 = a squaredP_1P_2; k squared =
[=x][=x]_1,

which determine P, [=x] and I or k. Amsler has constructed an integrator
which serves to determine these quantities by guiding a tracer once round
the boundary of the given figure (see below). Again, it may be required to
find the value of an integral [Integral]y[phi](x)dx between given limits
where [phi](x) is a simple function like sin nx, and where y is given as
the ordinate of a curve. The harmonic analysers described below are
examples of instruments for evaluating such integrals.

[Illustration: FIG. 19.]

[Illustration: FIG. 20.]

Amsler has modified his planimeter in such a manner that instead of the
area it gives the first or second moment of a figure about an axis in its
plane. An instrument giving all three quantities simultaneously is known as
Amsler's integrator or moment-planimeter. It has one tracer, but three
recording wheels. It is mounted on a [Sidenote: Amsler's Integrator.]
carriage which runs on a straight rail (fig. 19). This carries a horizontal
disk A, movable about a vertical axis Q. Slightly more than half the
circumference is circular with radius 2a, the other part with radius 3a.
Against these gear two disks, B and C, with radii a; their axes are fixed
in the carriage. From the disk A extends to the left a rod OT of length l,
on which a recording wheel W is mounted. The disks B and C have also
recording wheels, W_1 and W_2, the axis of W_1 being perpendicular, that of
W_2 parallel to OT. If now T is guided round a figure F, O will move to and
fro in a straight line. This part is therefore a simple planimeter, in
which the one end of the arm moves in a straight line instead of in a
circular arc. Consequently, the "roll" of W will record the area of the