Y = ∫cxydx or Y = ∫ydx + const.
the function Y has to be determined from the condition
| dY dx | = y. |
Graphically y = f(x) is either given by a curve, or the graph of the equation is drawn: y, therefore, and similarly Y, is a length. But dY/dx is in this case a mere number, and cannot equal a length y. Hence we introduce an arbitrary constant length a, the unit to which the integraph draws the curve, and write
| dY dx | = | y a | and aY = ∫ydx |
Now for the Y-curve dY/dx = tan φ, where φ is the angle between the tangent to the curve, and the axis of x. Our condition therefore becomes
| tan φ = | y a. |
This φ is easily constructed for any given point on the y-curve:—From the foot B′ (fig. 21) of the ordinate y = B′B set off, as in the figure, B′D = a, then angle BDB′ = φ. Let now DB′ with a perpendicular B′B move along the axis of x, whilst B follows the y-curve, then a pen P on B′B will describe the Y-curve provided it moves at every moment in a direction 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 Zürich, 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 φ, 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.