parallel to BD. The object of the integraph is to draw this new curve when
the tracer of the instrument is guided along the y-curve.

The first to describe such instruments was Abdank-Abakanowicz, who in 1889
published a book in which a variety of mechanisms to obtain the object in
question are described. Some years later G. Coradi, in Zuerich, carried out
his ideas. Before this was done, C.V. Boys, without knowing of
Abdank-Abakanowicz's work, actually made an integraph which was exhibited
at the Physical Society in 1881. Both make use of a sharp edge wheel. Such
a wheel will not slip sideways; it will roll forwards along the line in
which its plane intersects the plane of the paper, and while rolling will
be able to turn gradually about its point of contact. If then the angle
between its direction of rolling and the x-axis be always equal to [phi],
the wheel will roll along the Y-curve required. The axis of x is fixed only
in direction; shifting it parallel to itself adds a constant to Y, and this
gives the arbitrary constant of integration.

In fact, if Y shall vanish for x = c, or if

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 th