that is my trouble:
Let $\alpha : I \rightarrow R^3 $ a regular curve parameterized by its arc lenght, that is contained in the unit sphere.
If $k(s)>0$ and $\tau(s)\neq 0$ , $\forall s \in I$ then
$$ \frac{1}{k^2(s)} + \frac{(k'(s))^2}{k^4(s) \tau^2(s)} = 1 $$
my efforts were useless
We may easily generalize to the case in which $\alpha(s)$ lies in the sphere of radius $R$ centered at some point $C \in \Bbb R^3$, that is, to
$(\alpha(s) - C) \cdot (\alpha(s) - C) = R^2. \tag 1$
The key idea here is to express $\alpha(s) - C$ itself in terms of $T(s)$, $N(s)$, and $B(s)$, the Frenet frame of the curve $\alpha(s)$. We obtain, upon differentiating (1) with respect to the arc-length $s$,
$(\alpha(s) - C)' \cdot (\alpha(s) - C) = 0, \tag 2$
or
$\dot \alpha(s) \cdot (\alpha(s) - C) = 0; \tag 3$
since
$\dot \alpha(s) = T(s), \tag 4$
(3) becomes
$T(s) \cdot (\alpha(s) - C) = 0, \tag 5$
which shows that the $T(s)$ component of $\alpha(s) - C$ is zero. We next differentiate (5):
$\dot T(s) \cdot (\alpha(s) - C) + T(s) \cdot T(s) = 0, \tag 6$
and using the first Frenet-Serret equation
$\dot T(s) = \kappa(s) N(s) \tag 7$
and the fact that $T(s)$ is a unit vector,
$T(s) \cdot T(s) = 1, \tag 8$
we transform (6) into
$\kappa(s) N(s) \cdot (\alpha(s) - C) + 1 = 0, \tag 9$
and readily express the $N(s)$ component of $\alpha(s) - C$ as
$N(s) \cdot (\alpha(s) - C) = -\dfrac{1}{\kappa(s)}; \tag{10}$
taking the $s$ derivative of this equation yields
$\dot N(s) \cdot (\alpha(s) - C) + N(s) \cdot T(s) = \dfrac{\dot \kappa(s)}{\kappa^2(s)}; \tag{11}$
since
$N(s) \cdot T(s) = 0, \tag{12}$
and the second Frenet-Serret equation affirms that
$\dot N(s) = -\kappa(s) T(s) + \tau(s) B(s), \tag{13}$
we may now write (11) in the form
$(-\kappa(s) T(s) + \tau(s) B(s)) \cdot (\alpha(s) - C) = \dfrac{\dot \kappa(s)}{\kappa^2(s)}, \tag{14}$
and so by virtue of (5),
$ \tau(s) B(s) \cdot (\alpha(s) - C) = \dfrac{\dot \kappa(s)}{\kappa^2(s)}, \tag{15}$
immediately leading to
$B(s) \cdot (\alpha(s) - C) = \dfrac{\dot \kappa(s)}{\kappa^2(s) \tau(s)}, \tag{16}$
the $B(s)$ component of $\alpha(s) - C$; we combine (5), (10) and (16) and arrive at
$\alpha(s) - C = -\dfrac{1}{\kappa(s)} N(s) + \dfrac{\dot \kappa(s)}{\kappa^2(s) \tau(s)} B(s), \tag{17}$
now using
$N(s) \cdot B(s) = 0, \tag{18}$
it follows from (1) and (17) that
$R^2 = (\alpha(s) - C) \cdot (\alpha(s) - C) = \dfrac{1}{\kappa^2(s)} + \dfrac{(\dot \kappa(s))^2}{\kappa^4(s) \tau^2(s)}, \tag{19}$
the requested formula; we observe that when $R = 1$ we obtain the exact relationship given in the text of the question itself. $OE\Delta$.
See also this question.
