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 ⌠ 1 │ ———— _dy_ = log _y_, │ - ———— _d_(log _y_) = _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 ⌠ 1 _y_ = │ ————— _dt_ ⌡ _x_ 1 Then _Dₜy_ = —————, _D{_y} t_ = _x_, _x_