Y = [Integral,c:x]ydx,
then the axis of x has to be drawn through that point on the y-curve which
corresponds to x = c.
[Illustration: FIG. 22.]
In Coradi's integraph a rectangular frame F_1F_2F_3F_4 (fig. 22) rests with
four rollers R on the drawing board, and can roll freely in the direction
OX, which will be called the axis of the instrument. On the front edge
F_1F_2 travels a carriage AA' supported at A' on another rail. A bar DB can
turn about D, fixed to the frame in its axis, and slide through a point B
fixed in the carriage AA'. Along it a block K can slide. On the back edge
F_3F_4 of the frame another carriage C travels. It holds a vertical spindle
with the knife-edge wheel at the bottom. At right angles to the plane of
the wheel, the spindle has an arm GH, which is kept parallel to a [v.04
p.0980] similar arm attached to K perpendicular to DB. The plane of the
knife-edge wheel r is therefore always parallel to DB. If now the point B
is made to follow a curve whose y is measured from OX, we have in the
triangle BDB', with the angle [phi] at D,
tan [phi] = y/a,
where a = DB' is the constant base to which the instrument works. The point
of contact of the wheel r or any point of the carriage C will therefore
always move in a direction making an angle [phi] with the axis of x, whilst
it moves in the x-direction through the same distance as the point B on the
y-curve--that is to say, it will trace out the integral curve required, and
so will any point rigidly connected with the carriage C. A pen P attached
to this carriage will therefore draw the integral curve. Instead of moving
B along the y-curve, a tracer T fixed to the carriage A is guided along it.
For using the instrument the carriage is placed on the drawing-board with
the front edge parallel to the axis of y, the carriage A being clamped in
the central position with A at E and B at B' on the axis of x. The tracer
is then placed on the x-axis of the y-curve and clamped to the carriage,
and the instrument is ready for use. As it is convenient to have the
integral curve placed directly opposite to the y-curve so that
corresponding values of y or Y are drawn on the same line, a pen P' is
fixed to C in a line with the tracer.
Boys' integraph was invented during a sleepless night, and during the
following days carried out as a working model, which gives highly
satisfactory results. It is ingenious in its simplicity, and a direct
realization as a mechanism of the principles explained in connexion with
fig. 21. The line B'B is represented by the edge of an ordinary T-square
sliding against the edge of a drawing-board. The points B and P are
connected by two rods BE and EP, jointed at E. At B, E and P are small
pulleys of equal diameters. Over these an endless string runs, ensuring
that the pulleys at B and P always turn through equal angles. The pulley at
B is fixed to a rod which passes through the point D, which itself is fixed
in the T-square. The pulley at P carries the knife-edge wheel. If then B
and P are kept on the edge of the T-square, and B is guided along the
curve, the wheel at P will roll along the Y-curve, it having been
originally set parallel to BD. To give the wheel at P sufficient grip on
the paper, a small loaded three-wheeled carriage, the knife-edge wheel P
being one of its wheels, is added. If a piece of copying paper is inserted
between the wheel P and the drawing paper the Y-curve is drawn very
sharply.
Integraphs have also been constructed, by aid of which ordinary
differential equations, especially linear ones, can be solved, the solution
being given as a curve. The first suggestion in this direction was made by
Lord Kelvin. So far no really useful instrument has been made, although the
ideas seem sufficiently developed to enable a skilful instrument-maker to
produce one should there be sufficient demand for it. Sometimes a
combination of graphical work with an integraph will serve the purpose.
This is the case if the variables are separated, hence if the equation