The part of the instrument that draws the curve is a three-wheeled cart of lead, whose front wheel, F, is mounted, not as a caster, but like the steering wheel of a bicycle. When such a cart is moved, the front wheel, F, can only move in the direction of its own plane, whatever be the position of the cart; if, therefore, the cart is so moved that F is in the line, ee, and at the same time has its plane parallel to the rod, AB, then F must necessarily describe the required curve, and if it is made to pass over a sheet of black tracing paper, the required curve will be _drawn_. The upper end of the T square is raised above the 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. Of the various curves exhibited which have been drawn by the machine, the following are of special physical interest. Given the inclined straight line y = cx, the machine draws the parabola y = cx² / 2. This is the path of a projectile, as the space fallen is as the area of the triangle between the inclined line, the axis of x, and the traveling ordinate. Given the curve representing attraction y = 1 / x² the machine draws the hyperbola y = 1 / x the curve representing potential, as the work done in bringing a unit from an infinite distance to a point is measured by the area between the curve of attraction, the axis of x, and the ordinate at that point.