fulfilled, but the mobility of the new $(n+1)$–pod is zero.

On the contrary, condition (ii) is sufficient for the existence of a self–motion. If the platform is located in a way that the carrier line of $p_1, \ldots, p_m$ coincides with the carrier line of $P_{m+1}, \ldots, P_n$, then the platform can rotate freely about this line. Therefore we get a 1–dimensional set of so–called [*butterfly–motions*]{}.

Note that $n$–pods which fulfill the condition (ii) can also have further self–motions beside these butterfly–motions, even if they do not possess the property of item (i). Good examples for this fact are the three types of Bircard’s flexible octahedra [@bricardoct], as they can be interpreted as hexapods.

[**Result 2**]{}

*Let $\Pi$ be an $n$–pod with mobility $2$ or higher. Then one of the following holds:*

-   there are infinitely many pairs $(L,R)$ of elements of $S^2$ such that the points $\pi_L(p_1), \ldots, \pi_L(p_n)$ and $\pi_R(P_1), \ldots, \pi_R(P_n)$ are equivalent by an inversion or a similarity;

-   there exists $m \leq n$ such that $p_1, \ldots, p_m$ are collinear and $P_{m+1} = \ldots = P_n$, up to permutation of indices and interchange between base and platform;

-   there exists $m \leq n$ with $1 