The osculating sphere. 
Let $c$ be a Frenet curve in $\mathbb{R^3}$ with $\tau(s_0)\neq 0$. Then the surface of the sphere centred at point
$$c(s_0) + \frac{1}{\kappa(s_0)}e_2(s_0) - \frac{\kappa'(s_0)}{\tau(s_0)\kappa^2(s_0)}e_3(s_0)$$
which passes through the point $c(s_0)$, has point of contact with the curve at the point $s_0$ of the third order. This sphere is uniquely determined by these properties and is called the osculating sphere.

I would like to understand a certain part of the proof. 
Let
$$m(s_0) = c(s_0) + \alpha e_1(s_0) +\beta e_2(s_0)+ \gamma e_3(s_0)$$
be the centre of this hypothetical sphere.
Now according to my book $r^{(i)}(s) = 0,\quad i = 1,2,3$ where 
$$r(s) = \langle m-c(s), m-c(s) \rangle$$
This is do not understand.
I started to consider $(m-c(s))-(m-S(s))=S(s) - c(s)$ where $S(s)$ in the arc created by the intersection of the line between the center of the sphere and $c(s)$ by the sphere and found that 
$$(S-c)(s) \text{ has order of contact 3 at } s = s_0 \iff (S -c)^{(i)}(s_0) = 0 i = 1,2,3$$
Now I feel like I am nearly there. 

Could someone show why 
  $r^{(i)}(s) = 0, i = 1,2,3$
   is true mathematically?

 A: $\newcommand{\Reals}{\mathbf{R}}\newcommand{\Del}{\nabla}\newcommand{\eps}{\varepsilon}\newcommand{\Brak}[1]{\left\langle #1\right\rangle}$If $g$ and $h$ are smooth, real-valued functions defined in some open interval $(s_{0} - \eps, s_{0} + \eps)$ and if $N \geq 0$ is an integer, we say $g$ and $h$ have order-$N$ contact at $s_{0}$ if
$$
g^{(i)}(s_{0}) = h^{(i)}(s_{0}),\quad 0 \leq i \leq N.
$$
This is a condition on mappings, not on geometric objects. To define "order-$N$ contact" of a curve and a hypersurface (such as a sphere), I assume you're using a definition of this type:
Let $f:\Reals^{n} \to \Reals$ be smooth, and assume $\Del f$ is non-vanishing on the level set $M_{c} := \{f = 0\}$, so that $M_{0} \subset \Reals^{n}$ is a smooth hypersurface.
A smooth curve $c:(s_{0} - \eps, s_{0} + \eps) \to \Reals^{n}$ has order-$N$ contact with $M_{0}$ at $c(s_{0})$ if the one-variable function $f \circ c:(s_{0} - \eps, s_{0} + \eps) \to \Reals$ vanishes to order $N$ at $s_{0}$, i.e., if
$$
(f \circ c)^{(i)}(s_{0}) = 0,\quad 0 \leq i \leq N.
\tag{1}
$$
It may be necessary (or desirable) to assume in addition that $c$ is "regular to order $N$" in the sense that $c^{(i)}(s_{0}) \neq 0$ for $1 \leq i \leq N$.

Assuming this framework suits your purposes, let $m$ be a point of $\Reals^{3}$, let $c$ be a smooth path, and let
$$
f(x) = \Brak{m - x, m - x} = \|m - x\|^{2}.
$$
The sphere of radius $\|m - c(s_{0})\|$ centered at $m$ has order-$3$ contact with $c$ at $s_{0}$ if and only if
$$
r(s) = (f \circ c)(s) = \Brak{m - c(s), m - c(s)} = \|m - c(s)\|^{2}
$$
equals $r(s_{0}) = \|m - c(s_{0})\|^{2}$ to order three at $s_{0}$. By (1), this means
$$
r^{(i)}(s_{0}) = 0,\quad i = 1, 2, 3.
$$
