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?