HOW THE DIFFERENTIAL ANALYZER CALCULATES
Up to this point in this chapter, the author has tried to tell the story of the differential analyzer in plain words. But for reading this section, a little knowledge of calculus is necessary. ([See also Supplement 2].) If you wish, skip this section, and go on to the next one.
We have described how varying quantities, or variables, are operated on in the machine in one way or another: adding, subtracting, multiplying by a constant, referring to a table, and integrating. What do we do if we wish to multiply 2 variables together? A neat trick is to use the formula:
| xy = | ∫ x dy + | ∫ y dx |
To multiply in this way requires 2 integrators and 1 adder. The connections that are made between them are as follows:
| Shaft x | To Integrator 1, Screw |
| Shaft x | To Integrator 2, Disc |
| Shaft y | To Integrator 1, Disc |
| Shaft y | To Integrator 2, Screw |
| Integrator 1, Wheel | To Adder 1, Input 1 |
| Integrator 2, Wheel | To Adder 1, Input 2 |
| Adder 1, Output | To Shaft expressing xy |
A product of 2 variables under the integral sign can be obtained a little more easily, because of the curious powers of the differential analyzer. Thus, if it is desired to obtain ∫ xy dt, we can use the formula:
| xy dt = | x d | y dt |
and this operation does not require an adder. The connections are as follows:
| Shaft t | To Integrator 1, Disc |
| Shaft y | To Integrator 1, Screw |
| Integrator 1, Wheel | To Integrator 2, Disc |
| Shaft x | To Integrator 2, Screw |
| Integrator 2, Wheel | To Shaft expressing ∫xy dt |
In order to get the quotient of 2 variables, x/y, we can use some more tricks. First, the reciprocal 1/y can be obtained by using the two simultaneous equations:
 | 1 | dy = log y, | |
| y | |||
| - | 1 | d(log y) = y |
| y |
The connections are as follows:
| Shaft y | To Integrator 1, Disc and to Integrator 2, Wheel |
| Shaft log y | To Integrator 1, Wheel and to Integrator 2, Disc |
| Shaft 1/y | To Integrator 1, Screw, and negatively to Integrator 2, Screw |
In order to get x/y, we can then multiply x by 1/y. We see that this setup gives us log y for nothing, that is, without needing more integrators or other equipment. Clearly, other tricks like this will give sin x, cos x, eˣ, x², and other functions that satisfy simple differential equations.
An integral of a reciprocal can be obtained even more directly. Suppose that
| y = | | 1 | dt |
| x |
Then
| Dₜ y = | 1 | , Dy t = x, |
| x |
| t = | x dy |
The connections therefore are:
| Shaft t | To Integrator, Wheel |
| Shaft x | To Integrator, Screw |
| Shaft y | To Integrator, Disc |
The light wheel then drives the heavy disc. Clearly only the angle-indicator device makes this possible at all. Naturally, the closer the wheel gets to the center of the disc, that is, x approaching zero, the greater the strain on the mechanism, and the more likely the result is to be off. Mathematically, of course, the limit of 1/x as x approaches zero equals infinity, and this gives trouble in the machine.
There is no standard mathematical method for solving any differential equation. But the machine provides a standard direct method for solving all differential equations with only one independent variable. First: assign a shaft for each term that appears in the equation. For example, the highest derivative that appears and the independent variable are both assigned shafts. The integral of the highest derivative is easily obtained, and the integral of that integral, etc. Second: connect the shafts so that all the mathematical relations are expressed. Both explicit and implicit equations may be expressed. Third: for any shaft there must be just one drive, or source of torque. A shaft may, however, drive more than one other shaft. Fourth: choose scale factors so that the limits of the machine are not exceeded yet at the same time are well used. For example, the most that an integrator or a function table can move is 1 or 2 feet. Also, the number of full turns made by a shaft in representing its variable should be large, often between 1000 and 10,000.
Of course, as with all these large machines, anyone would need some months of actual practice before he could put on a problem and get an answer efficiently.