$y^{2} = x$ , satisfied by pairs of such numbers. Then whatever integral value is given to $y$ , $x$ can assume one corresponding integral value. So the "field" for $y$ is unrestricted among these positive or negative integers. But the "field" for $x$ is restricted in two ways. In the first place $x$ must be positive, and in the second place, since $y$ is to be integral, $x$ must be a perfect square. Accordingly, the "field" of $x$ is restricted to the set of integers $1^{2}$ , $2^{2}$ , $3^{2}$ , $4^{2}$ , and so on, i.e., to $1$ , $4$ , $9$ , $16$ , and so on.

The study of the general properties of a relation between pairs of numbers is much facilitated by the use of a diagram constructed as follows: 1

Draw two lines $O X$ and $O Y$ at right angles; let any number $x$ be represented by $x$ units

(in any scale) of length along $O X$ , any number $y$ by $y$ units (in any scale) of length along $O Y$ . Thus if $O M$ , along $O X$ , be $x$ units in length, and $O N$ , along $O Y$ , be $y$ units in length, by completing the parallelogram $O M P N$ we find a point $P$ which corresponds to the pair of numbers $x$ and $y$ . To each point there corresponds one pair of numbers, and to each pair of numbers there corresponds one point. The pair of numbers are called the coordinates of the point. Then the points whose coordinates satisfy some fixed relation can be indicated in a convenient way, by drawing a line, if they all lie on a line, or by shading an area if they are all points in the area. If the relation can be represented by an equation such as $x + y = 1$ , or $y^{2} = x$ , then the points lie on a line, which is straight in the former case and curved in the latter. For example, considering only positive numbers, the points whose coordinates satisfy $x + y = 1$ lie on the straight line $A B$ in 1, where $0 A = 1$ and $O B = 1$ . Thus this segment of the straight line $A B$ gives a pictorial representation of the properties of the relation under the restriction to positive numbers.

Another example of a relation between two variables is afforded by considering the variations in the pressure and volume of a given mass of some gaseous substance–-such as air

or coal-gas or steam–-at a constant temperature. Let $v$ be the number of cubic feet in its volume and $p$ its pressure in lb. weight per square inch. Then the law, known as Boyle's law, expressing the relation between $p$ and $v$ as both vary, is that the product $p v$ is constant, always supposing that the temperature does not alter. Let us suppose, for example, that the quantity of the gas and its other circumstances are such that we can put $p v = 1$ (the exact number on the right-hand side of the equation makes no essential difference). 2

Then in 2 we take two lines, $O V$ and $O P$ , at right angles and draw $O M$ along $O V$ to represent $v$ units of volume, and $O N$ along $O P$

to represent $p$ units of pressure. Then the point $Q$ , which is found by completing the parallelogram $O M Q N$ , represents the state of the gas when its volume is $v$ cubic feet and its pressure is $p$ lb. weight per square inch. If the circumstances of the portion of gas considered are such that $p v = 1$ , then all these points $Q$ which correspond to any possible state of this portion of gas must lie on the curved line $A B C$ , which includes all points for which $p$ and $v$ are positive, and $p v = 1$ . 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 $Q$ must be near $C$ , or even beyond $C$ on the undrawn part of the curve; then the volume will be very small. When the volume is big $Q$ will be near to $A$ , or beyond $A$ ; 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 $p$ when