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 opposite 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 generates an area. If therefore a line OA = QT
is turning about a fixed point O, always keeping parallel to QT, it will
sweep over an area equal to that generated by the more general motion of
QT. Let now (fig. 17) QT be placed on OA, and T be guided round the closed
curve in the sense of the arrow. Q will describe a curve OSB. It may be
made visible by putting a piece of "copying paper" under the hatchet. When
T has returned to A the hatchet has the position BA. A line turning from OA
about O kept parallel to QT will describe the circular sector OAC, which is
equal in magnitude and sense to AOB. This therefore measures the area
generated by the motion of QT. To make this motion cyclical, suppose the
hatchet turned about A till Q comes from B to O. Hereby the sector AOB is
again described, and again in the positive sense, if it is remembered that
it turns about the tracer T fixed at A. The whole area now generated is
therefore twice the area of this sector, or equal to OA. OB, where OB is
measured along the arc. According to the theorem given above, this area
also equals the area of the given curve less the area OSBO. To make this
area disappear, a slight modification of the motion of QT is required. Let
the tracer T be moved, both from the first position OA and the last BA of
the rod, along some straight line AX. Q describes curves OF and BH
respectively. Now begin the motion with T at some point R on AX, and move
it along this line to A, round the curve and back to R. Q will describe the
curve DOSBED, if the motion is again made cyclical by turning QT with T
fixed at A. If R is properly selected, the path of Q will cut itself, and
parts of the area will be positive, parts negative, as marked in the
figure, and may therefore be made to vanish. When this is done the area of
the curve will equal twice the area of the sector RDE. It is therefore
equal to the arc DE multiplied by the length QT; if the latter equals 10
in., then 10 times the number of inches contained in the arc DE gives the
number of square inches contained within the given figure. If the area is
not too large, the arc DE may be replaced by the straight line DE.
To use this simple instrument as a planimeter requires the possibility of
selecting the point R. The geometrical theory here given has so far failed
to give any rule. In fact, every line through any point in the curve
contains such a point. The analytical theory of the inventor, which is very
similar 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° 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 ½l,
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, 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 = Ak² 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 = a²P_1P_2; k² =
[=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
figure. Imagine now that the disks B and C also receive arms of length l
from the centres of the disks to points T_1 and T_2, and in the direction
of the axes of the wheels. Then these arms with their wheels will again be
planimeters. As T is guided round the given figure F, these points T_1 and
T_2 will describe closed curves, F_1 and F_2, and the "rolls" of W_1 and
W_2 will give their areas A_1 and A_2. Let XX (fig. 20) denote the line,
parallel to the rail, on which O moves; then when T lies on this line, the
arm BT_1 is perpendicular to XX, and CT_2 parallel to it. If OT is turned
through an angle [theta], clockwise, BT_1 will turn counter-clockwise
through an angle 2[theta], and CT_2 through an angle 3[theta], also
counter-clockwise. If in this position T is moved through a distance x
parallel to the axis XX, the points T_1 and T_2 will move parallel to it
through an equal distance. If now the first arm is turned through a small
angle d[theta], moved back through a distance x, and lastly turned back
through the angle d[theta], the tracer T will have described the boundary
of a small strip of area. We divide the given figure into [v.04 p.0979]
such strips. Then to every such strip will correspond a strip of equal
length x of the figures described by T_1 and T_2.
The distances of the points, T, T_1, T_2, from the axis XX may be called y,
y_1, y_2. They have the values
y = l sin [theta], y_1 = l cos 2[theta], y_2 = -l sin 3[theta],
from which
dy = l cos [theta].d[theta], dy_1 = - 2l sin 2[theta].d[theta], dy_2 = -
3l cos 3[theta].d[theta].