In § 20 we considered two variables connected by a relation. We may similarly consider three variables (\(x\), \(y\), and \(z\)) connected by a relation such that when the values of \(x\) and \(y\) are both given, the value or values of \(z\) are known. In this case we call \(z\) a function of the two variables \(x\) and \(y\); \(x\) and \(y\) the independent variables, \(z\) the dependent variable; and we express this dependence of \(z\) upon \(x\) and \(y\) by writing \[z = f(x, y).\] The remarks of § 20 may all be applied, mutatis mutandis, to this more complicated case.
The method of representing such functions of two variables graphically is exactly the same in principle as in the case of functions of a single variable. We must take three axes, \(OX\), \(OY\), \(OZ\) in space of three dimensions, each axis being perpendicular to the other two. The point \((a, b, c)\) is the point whose distances from the planes \(YOZ\), \(ZOX\), \(XOY\), measured parallel to \(OX\), \(OY\), \(OZ\), are \(a\), \(b\), and \(c\). Regard must of course be paid to sign, lengths measured in the directions \(OX\), \(OY\), \(OZ\) being regarded as positive. The definitions of coordinates, axes, origin are the same as before.
Now let \[z = f(x, y).\] As \(x\) and \(y\) vary, the point \((x, y, z)\) will move in space. The aggregate of all the positions it assumes is called the locus of the point \((x, y, z)\) or the graph of the function \(z = f(x, y)\). When the relation between \(x\), \(y\), and \(z\) which defines \(z\) can be expressed in an analytical formula, this formula is called the equation of the locus. It is easy to show, for example, that the equation \[Ax + By + Cz + D = 0\] (the general equation of the first degree) represents a plane, and that the equation of any plane is of this form. The equation \[(x – \alpha)^{2} + (y – \beta)^{2} + (z – \gamma)^{2} = \rho^{2},\] or \[x^{2} + y^{2} + z^{2} + 2Fx + 2Gy + 2Hz + C = 0,\] where \(F^{2} + G^{2} + H^{2} – C > 0\), represents a sphere; and so on. For proofs of these propositions we must again refer to text-books of Analytical Geometry.
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