**C. The contact of plane curves.** Two curves are said to *intersect* (or *cut*) at a point if the point lies on each of them. They are said to *touch* at the point if they have the same tangent at the point.

Let us suppose now that \(f(x)\), \(\phi(x)\) are two functions which possess derivatives of all orders continuous for \(x = \xi\), and let us consider the curves \(y = f(x)\), \(y = \phi(x)\). In general \(f(\xi)\) and \(\phi(\xi)\) will not be equal. In this case the abscissa \(x = \xi\) does not correspond to a point of intersection of the curves. If however \(f(\xi) = \phi(\xi)\), the curves intersect in the point \(x = \xi\), \(y = f(\xi) = \phi(\xi)\). Let us suppose this to be the case. Then in order that the curves should not only cut but touch at this point it is obviously necessary and sufficient that the first derivatives \(f'(x)\), \(\phi'(x)\) should also have the same value when \(x = \xi\).

The contact of the curves in this case may be regarded from a different point of view. In the figure the two curves are drawn touching at \(P\), and \(QR\) is equal to \(\phi(\xi + h) – f(\xi + h)\), or, since \(\phi(\xi) = f(\xi)\), \(\phi'(\xi) = f'(\xi)\), to \[\tfrac{1}{2} h^{2}\{\phi”(\xi + \theta h) – f”(\xi + \theta h)\},\] where \(\theta\) lies between \(0\) and \(1\). Hence \[\lim \frac{QR}{h^{2}} = \tfrac{1}{2}\{\phi”(\xi) – f”(\xi)\},\] when \(h \to 0\). In other words, when the curves touch at the point whose abscissa is \(\xi\), *the difference of their ordinates at the point whose abscissa is \(\xi + h\) is at least of the second order of smallness when \(h\) is small*.

The reader will easily verify that \(\lim (QR/h) = \phi'(\xi) – f'(\xi)\) when the curves cut and do not touch, so that \(QR\) is then of the first order of smallness only.

It is evident that the degree of smallness of \(QR\) may be taken as a kind of measure of the *closeness of the contact* of the curves. It is at once suggested that if the first \(n – 1\) derivatives of \(f\) and \(\phi\) have equal values when \(x = \xi\), then \(QR\) will be of \(n\)th order of smallness; and the reader will have no difficulty in proving that this is so and that \[\lim \frac{QR}{h^{n}} = \frac{1}{n!}\{\phi^{(n)}(\xi) – f^{(n)}(\xi)\}.\] We are therefore led to frame the following definition:

**Contact of the \(n\)th order.** *If \(f(\xi) = \phi(\xi)\), \(f'(\xi) = \phi'(\xi)\), …, \(f^{(n)}(\xi) = \phi^{(n)}(\xi)\), but \(f^{(n+1)}(\xi) \neq \phi^{(n+1)}(\xi)\), then the curves \(y = f(x)\), \(y = \phi(x)\) will be said to have contact of the \(n\)th order at the point whose abscissa is \(\xi\).*

The preceding discussion makes the notion of contact of the \(n\)th order dependent on the choice of axes, and fails entirely when the tangent to the curves is parallel to the axis of \(y\). We can deal with this case by taking \(y\) as the independent and \(x\) as the dependent variable. It is better, however, to consider \(x\) and \(y\) as functions of a parameter \(t\). An excellent account of the theory will be found in Mr Fowler’s tract referred to on p. 266, or in de la Vallée Poussin’s *Cours d’Analyse*, vol. ii, pp. 396 *et seq.*

- A much fuller discussion of the theory of curvature will be found in Mr Fowler’s tract referred to on sec. 148.↩︎

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