We have hitherto used the notation \[\begin{equation*} y = f(x) \tag{1}\end{equation*}\] to express functional dependence of \(y\) upon \(x\). It is evident that this notation is most appropriate in the case in which \(y\) is expressed explicitly in terms of \(x\) by means of a formula, as when for example \[y = x^{2},\quad \sin x,\quad a\cos^{2}x + b\sin^{2}x.\]

We have however very often to deal with functional relations which it is impossible or inconvenient to express in this form. If, for example, \(y^{5} – y – x = 0\) or \(x^{5} + y^{5} – ay = 0\), it is known to be impossible to express \(y\) explicitly as an algebraical function of \(x\). If \[x^{2} + y^{2} + 2Gx + 2Fy+ C = 0,\] \(y\) can indeed be so expressed, viz. by the formula \[y = -F + \sqrt{F^{2} – x^{2} – 2Gx – C};\] but the functional dependence of \(y\) upon \(x\) is better and more simply expressed by the original equation.

It will be observed that in these two cases the functional relation is fully expressed by equating a function of the two variables \(x\) and \(y\) to zero,  by means of an equation \[\begin{equation*} f(x, y) = 0. \tag{2}\end{equation*}\]

We shall adopt this equation as the standard method of expressing the functional relation. It includes the equation  as a special case, since \(y – f(x)\) is a special form of a function of \(x\) and \(y\). We can then speak of the locus of the point \((x, y)\) subject to \(f(x, y) = 0\), the graph of the function \(y\) defined by \(f(x, y) = 0\), the curve or locus \(f(x, y) = 0\), and the equation of this curve or locus.

There is another method of representing curves which is often useful. Suppose that \(x\) and \(y\) are both functions of a third variable \(t\), which is to be regarded as essentially auxiliary and devoid of any particular geometrical significance. We may write \[\begin{equation*} x = f(t),\quad y = F(t). \tag {3}\end{equation*}\] If a particular value is assigned to \(t\), the corresponding values of \(x\) and of \(y\) are known. Each pair of such values defines a point \((x, y)\). If we construct all the points which correspond in this way to different values of \(t\), we obtain the graph of the locus defined by the equations . Suppose for example \[x = a\cos t,\quad y = a\sin t.\] Let \(t\) vary from \(0\) to \(2\pi\). Then it is easy to see that the point \((x, y)\) describes the circle whose centre is the origin and whose radius is \(a\). If \(t\) varies beyond these limits, \((x, y)\) describes the circle over and over again. We can in this case at once obtain a direct relation between \(x\) and \(y\) by squaring and adding: we find that \(x^{2} + y^{2} = a^{2}\), \(t\) being now eliminated.

Examples XVIII

1. The points of intersection of the two curves whose equations are \(f(x, y) = 0\), \(\phi(x, y) = 0\), where \(f\) and \(\phi\) are polynomials, can be determined if these equations can be solved as a pair of simultaneous equations in \(x\) and \(y\). The solution generally consists of a finite number of pairs of values of \(x\) and \(y\). The two equations therefore generally represent a finite number of isolated points.

2. Trace the curves \((x + y)^{2} = 1\), \(xy = 1\), \(x^{2} – y^{2} = 1\).

3. The curve \(f(x, y) + \lambda\phi(x, y) = 0\) represents a curve passing through the points of intersection of \(f = 0\) and \(\phi = 0\).

4. What loci are represented by \[(\alpha) x = at + b,\quad y = ct + d,\qquad (\beta)  x/a = 2t/(1 + t^{2}),\quad y/a = (1 – t^{2})/(1 + t^{2}),\] when \(t\) varies through all real values?

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