Page:IntroductionToMathematicsWhitehead.pdf/27

${\displaystyle OP}$ to represent ${\displaystyle p}$ units of pressure. Then the point ${\displaystyle Q}$, which is found by completing the parallelogram ${\displaystyle MONQ}$, represents the state of the gas when its volume is ${\displaystyle v}$ cubic feet and its pressure is ${\displaystyle p}$ lb. weight per square inch. If the circumstances of the portion of gas considered are such that ${\displaystyle pv=1}$, then all these points ${\displaystyle Q}$ which correspond to any possible state of this portion of gas must lie on the curved line ${\displaystyle ABC}$, which includes all points for which ${\displaystyle p}$ and ${\displaystyle v}$ are positive, and ${\displaystyle pv=1}$. Thus this curved line gives a pictorial representation of the relation holding between the volume and the pressure. When the pressure is very big the corresponding point ${\displaystyle Q}$ must be near ${\displaystyle C}$, or even beyond ${\displaystyle C}$ on the undrawn part of the curve; then the volume will be very small. When the volume is big ${\displaystyle Q}$ will be near to ${\displaystyle A}$, or beyond ${\displaystyle A}$; and then the pressure will be small. Notice that an engineer or a physicist may want to know the particular pressure corresponding to some definitely assigned volume. Then we have the case of determining the unknown ${\displaystyle p}$ when ${\displaystyle v}$ is a known number. But this is only in particular cases. In considering generally the properties of the gas and how it will be have, he has to have in his mind the general form of the whole curve ${\displaystyle ABC}$ and its general properties. In other words the really fundamental idea is that of the pair of variables