FUNCTIONALITY
If we pour some mercury into a U-tube closed at one end, the air in this end will be contained in a closed vessel under pressure. We can increase the pressure by pouring more mercury into the open end of the tube. We can measure the volume of the air by measuring the length of the tube which it occupies. We can measure the pressure on this air by measuring the difference of length of the mercury in the two limbs of the tube. By taking all necessary precautions we shall find that for each value which the pressure attains there is a corresponding value of the volume of the air.
Fig. 27.
We thus find the pressure values, p1, p2, p3, p4, p5, etc., and the corresponding volumes, v1, v2, v3, v4, v5, etc., and we may then plot these values so as to make a graph.
In this figure the values represented along the horizontal axis are pressure-values, and those represented along the vertical axis are volume-values. We have so made the experiment that we can make the pressure-values whatever we choose—let us call them the values of the independent variable or argument. For each value of the pressure, or argument, there is a corresponding value of the volume, which depends on the pressure—let us call these values of the volume values of the dependent variable or function.
We can make arbitrary values of the pressure, but whenever we do this the corresponding values of the volume are fixed. We say, then, that the volume is a function of the pressure. In general, when we choose one value of an independent variable, or argument, there can be only one, or a small number, of values of the dependent variable, or function. If there are two or more values of the function for one value of the argument each of these is necessarily determined by the value which we choose to assign to the argument. There is a strict functionality between the two series of variables. In the experiment we have chosen this functionality is expressed by the equation pv = k(1 + at), where p is the pressure, v the volume, k and a constants, and t is the temperature at which the experiment is carried out. In a number of experiments like that which we have mentioned, k, a, and t are the same throughout, and this is why we call them constants. We give p any value we like, and then v can be calculated from the equation.