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. We thus find the pressure values, _p_↓{1}, _p_↓{2}, _p_↓{3}, _p_↓{4}, _p_↓{5}, etc., and the corresponding volumes, _v_↓{1}, _v_↓{2}, _v_↓{3}, _v_↓{4}, _v_↓{5}, etc., and we may then plot these values so as to make a graph. [Illustration: FIG. 27.] 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. RATE OF VARIATION