PRODUCT INTEGRAPH. The product integraph belongs to that class of calculating machines which deals with curves and graphs rather than with numbers. It multiplies together any two given curves before the integration. Results are obtained by an automatically-drawn curve. It can be used, therefore, to determine the coefficients in the expansion of an arbitrary func tion into any chosen set of orthogonal functions. It can be extended to make the form of one or both of the curves to be integrated dependent upon the result of the integration by having the mechanism which operates the recording pencil also control the input curves as the operation of integration proceeds. The device is then capable of solving ordinary differential equations, and this is its most important use. The result of the first integra tion may be integrated again any number of times, thus raising the order of the equation solvable. In its most developed form, therefore, the product integraph is a machine for solving ordinary differential equations where the coefficients are given by means of curves. Hence the coefficients may be discontinuous or even multiple-valued without making the mechanical work any more involved, and the device allows the prompt solution of equations which are beyond formal treatment except by extremely labouri ous processes. The result given by the machine is in the form of a curve or family of curves, corresponding to numerically-chosen initial or boundary conditions.
The first product integraph constructed at the Massachusetts Institute of Technology is adapted for the solution of second order differential equations of nearly, but not quite, general form. It will treat equations of the form where each of fi and f, may be functions of any one of the quantities z, x or By By schemes of interpolation double-valued dx functions may be introduced, and any of the input functions may be functions of any two of the three quantities. Practically all of the second-order differential equations met in practice are therefore subject to treatment, with the reservation of course that graphical solutions, not formal solutions, are always obtained.
The solution of equations having discontinuous coefficients has proved especially interesting.
To solve an equation such as the above, the procedure is as f ollows: The functions fi and f, are plotted and mounted on movable platens. (See Plate I.) Operators keep a movable
pointer on each of the curves as the platens are moved by the machine. The ordinates given by the position of the pointers control the action of the device so that the necessary addition and multiplication are made, the two integrations performed, and the results plotted. The necessary operations are more clearly indicated when the equation is in the form:— the result curve thus being a plot of z as a function of x for chosen initial conditions a and b. The movement of the platens is con trolled by the device itself. When all input platens are driven by the same mechanism which drives the result platen, the func tions are evidently functions of x. A cross connection enables any input platen to be driven by the mechanism which operates the result pencil. Then an input function of z is being used. A drive can also be obtained from the result of the first integrations alone, giving an input function of dx In this first instrument, multiplication and first integration are performed by a Thomson watt-hour meter, the second integra tion being mechanical. Servo mechanisms do the work of moving the parts without imposing load on the delicate integrating devices. Precision of 1-2% is attained, depending upon many conditions.
In the second instrument, now partially in operation, all integration, addition and multiplication is mechanical, servo mechanisms and torque amplifiers being utilized to enable pre cision apparatus to operate at practically no load. A unit system is adopted so that the range of the instrument may be extended at will. At present sufficient parts are made available to handle sixth-order equations, or a set of three second-order equations. The precision is much higher than in the first instrument.
The first instrument and some of its applications are described in detail in the following publications: V. Bush, F. D. Gage and H. R. Stewart, "A Continuous Integraph," Jour. of the Franklin Inst. ( Jan. 1927) ; V. Bush and H. L. Hazen, "Integraph Solution of Differential Equations," Jour. of the Franklin Inst. (Nov. 1927) . Early work on this type of instrument is presented in a series of articles by Sir William and Professor James .Thompson, Royal Society Proceedings, Vol. 24, 1875-76. See also MATHEMATICAL INSTRUMENTS (V. Bu.)