All the integrating machines hitherto made, of which I can find any record, may be classed under two heads, one of which, Ainslee's machine, is the sole representative, depending on the revolution of a disk which partly rolls and partly slides on the paper, and the other comprising all the remaining machines depending on the varying diameters of the parts of a rolling system. Now, none of these machines do their work by the method of the mathematician, but in their own way. My machine, however, is an exact mechanical translation of the mathematical method of integrating y dx, and thus forms a third type of instrument.
The mathematical rule may be described in words as follows: Required the area between a curve, the axis of x and two ordinates; it is necessary to draw a new curve, such that its steepness, as measured by the tangent of the inclination, may be proportional to the ordinate of the given curve for the same value of x, then the ascent made by the new curve in passing from one ordinate to the other is a measure of the area required.
The figure shows a plan and side elevation of a model of the instrument, made merely to test the idea, and the arrangement of the details is not altogether convenient. The frame-work is a kind of T square, carrying a fixed center, B, which moves along the axis of x of the given curve, a rod passing always through B carries a pointer, A, which is constrained to move in the vertical line, ee, of the T square, A then may be made to follow any given curve. The distance of B from the edge, ee, is constant; call it K, therefore, the inclination of the rod, AB, is such that its tangent is equal to the ordinate of the given curve divided by K; that is, the tangent of the inclination is proportional to the ordinate; therefore, as the instrument is moved over the paper, AB has always the inclination of the desired curve.
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.
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.