Now let us watch this mechanism move. If the disc turns a little bit, the wheel pressing on it must turn a little bit. If the screw turns a small amount, the distance between the edge of the wheel and the center of the disc changes. The amount that the wheel turns is doubled if its distance from the center of the disc is doubled, and halved if that distance is halved. So we see that: (_the total amount that the wheel turns_) EQUALS THE SUM OF ALL THE SMALL (_amounts of turning_), EACH EQUAL TO: A BIT OF (_disc turning_) MULTIPLIED BY THE (_distance from the center of the disc to the edge of the wheel_) APPLYING TO THAT BIT If we look back at our discussion of integrating (p. 72), we see that the capital words here are just the same as those used there. Thus we have a mechanism that expresses integration: (_the total amount that the wheel turns_) EQUALS THE INTEGRAL OF (_the distance from the center of the disc to the wheel_) WITH RESPECT TO (_the amount that the disc turns_) The scheme of this mechanism is shown in Fig. 20. For example, suppose that the screw measures the speed at which a car travels and that the disc measures time. The wheel, consequently, will measure distance traveled by the car. The mechanism INTEGRATES speed with respect to time and gives distance. [Illustration: FIG. 20. Scheme of integrator.] This mechanism is the device that Lord Kelvin talked about in 1879 and that Dr. Bush made practical in 1925. The mechanical difficulty is to make the friction between the disc and the wheel turn the wheel with enough force to do other work. In the second differential analyzer, the angle indicator set on the shaft of the wheel solves the problem very neatly. [Illustration: FIG. 21. Graph of air resistance coefficient.] Function Tables The behavior of some physical quantities can be described only by a series of numbers or a graphic curve. For example, the _resistance_ or _drag_ of the air against a passing object is related to the speed of the object in a rather complicated way. Part of the relation is called the _drag coefficient_ or _resistance coefficient_; a rough graph of this is shown in Fig. 21. This graph shows several interesting facts: (1) when the object is still, there is no air resistance; (2) as it travels faster and faster, air resistance rapidly increases; (3) when the object travels with the speed of sound, resistance is very great and increases enormously; (4) but, when the object starts traveling with a speed about 20 per cent faster than sound, the drag coefficient begins to decrease. This drawing or _graph_ shows in part how air resistance depends on speed of object; in other words, it shows the drag coefficient as a _function_ of speed (see Supplement 2). [Illustration: FIG. 22. Pointer following graph.]