paper, and forms a bridge, under which the cart travels. There is a
longitudinal slot in this bridge in which lies a horizontal wheel,
carried by that part of the cart corresponding to the head of a bicycle.
By this means the horizontal motion communicated to the front wheel of
the cart by the bridge, is equal to that of the pointer, A; at the same
time the cart is free to move vertically.

The mechanism employed to keep the plane of the front wheel of the cart
parallel to AB is made clear by the figure. Three equal wheels at the
ends of two jointed arms are connected by an open band, as shown. Now,
in an arrangement of this kind, however the arms or the wheels are
turned, lines on the wheels, if ever parallel, will always be so. If,
therefore, the wheel at one end is so supported that its rotation is
equal to that of AB, while the wheel at the other end is carried by the
fork which supports F, then the plane of F, if ever parallel to AB, will
always be so. Therefore, when A is made to trace any given curve, F will
draw a curve whose ascent is (1/K) f y dx, and this, multiplied by K, is
the area required.

[Illustration: AN INTEGRATING MACHINE.]

Not only does the machine integrate y dx, but if the plane of the front
wheel of the cart is set at right angles instead of parallel to AB, then
the cart finds the integral of dx / y, and thus solves problems, such,
for instance, as the time occupied by a body in moving along a path when
the law of the velocity is known.

Some modifications of the machine already described will enable it to
integrate squares, cubes, or products of functions, or the reciprocals
of any of these.