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\section*{An example from Struik's
  \textit{Lectures on Classical Differential Geometry}}

This procedure is simply a generalization of the method used in Sects.
1-3 and 1-4 to obtain the equations of the osculating plane and the
osculating circle.  Let $f(u)$ near $P(u=u_0)$ have finite derivatives
$f^{(i)}(u_0)$, $i = 1, 2, \ldots, n+1$.  Then if we take $u=u_1$ at $A$
and write $h = u_1 - u_0$, then there exists a Taylor development of $f(u)$
of the form (compare Eq.  (1-5)):
$$
f(u_1) = f(u_0) + hf'(u_0)+{\frac{h^2}{2!}}f''(u_0) + \cdots
 + {\frac{h^{n+1}}{(n+1)!}}f^{(n+1)}(u_0) + o(h^{n+1}).
$$

Here, $f(u_0)=0$ since $P$ lies on $\Sigma_2$, and $h$ is of order $AP$
(see theorem Sec.  1-2); $f(u_1)$ is of order $AD$.  \textit{Hence necessary and
sufficient conditions that the surface has a contact of order $n$ at $P$
with the curve are that at $P$ the relations hold:}
\[
f(u) = f'(u) = f''(u) = \cdots = f^{(n)}(u) = 0;\quad f^{(n+1)}(u) \ne 0.
\]




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